Frequency comparator utilizing enveloping-event detection via symbolic dynamics of fixed or modulated waveforms

ABSTRACT

Systems, algorithms, circuits, and methods for pattern detection of signature events in signal dynamics defined by instantaneous states of applied square-wave signals. Selected patterns may be recognized individually or in equivalence classes, and detection may be implemented via state or transition analysis. Varieties of conditions may be detected in parallel, including phase, ambiguity states, and frequency comparison. One embodiment realizes a real-time frequency comparator for asynchronous square-wave signals. Realizations detect various classes of symmetry conditions unique to enveloping events occurring for these classes of square-wave signal pairs. This approach provides feedback-free implementations operating over an extremely wide frequency range and does not require signals of quadrature form. A typical logic circuit implementation involves two to four flip-flops, or two-stage two-bit shift registers, and modest combinational logic. The resulting system can be readily implemented as a utility integrated circuit of modest scale or as a small-scale “IP core” within larger system-on-a-chip (SoIC) devices.

CROSS REFERENCE TO RELATED APPLICATIONS

This Application claims benefit of priority from provisional patentapplication Ser. No. 60/707,287, filed Aug. 10, 2005.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention pertains to electronics, algorithms, signalprocessing, communications systems and computers, and more specificallyto frequency comparators, phase comparators, and processes involvingsymbolic dynamics and quasi-periodic phenomena.

2. Description of the Related Art

Frequency and phase comparators are classic subsystem elements used in awide variety of applications in communications, signal analysis, andother areas. A number of approaches have been employed or proposedincluding the use of digital counters, analog integrators,quadrature-phase signal formats provided in parallel, and state machineswith state feedback.

SUMMARY OF THE INVENTION

Embodiments of the present invention utilize “enveloping” eventphenomena, intrinsic to the dynamics of pairs of square waves or pulsewaves waveform signals of different frequencies, to determine which ofthe pair is of a higher frequency than the other, as well as other typesof information. The frequency and duty-cycle of one or both of thewaveforms may be modulated in time, allowing many applications,including those in communications and measurement instrumentation. Thepotential frequency operating range is enormous: the low end isultimately determined by state memory duration (i.e., waveform periodtime-scales of up to multiple years) and the high-end is ultimatelydetermined by waveform-transition detection recovery intervals (i.e.,waveform period time-scales of down to 2-3 logic gate propagationtimes).

An enveloping-event detection approach may be used to facilitate atleast two new classes of frequency comparator and related functions thatare entirely feed-forward in state signal flow, and unlike the relatedart, do not require digital counters, analog integrators,quadrature-phase signal formats, or state machines with state signalfeedback. These approaches work over an extremely wide frequency range.The enveloping-event detection and classification can be done in anumber of ways as provided for by embodiments of the invention.

Additionally, embodiments of the present invention may be applied tosynchronous motor control and operation, transportation systems,manufacturing or project scheduling, the scheduling of real-time tasksin operating systems, astronomy calculations (including analyses ofancient archaeoastronomy sites), oscillator-coupling phenomena inchaotic and self-organizing systems, geometric lattice design, quantumeffects, long-duration timing systems for radioactive waste storage orlong-distance space travel, molecular vibration, energy-transfer amonginharmonic periodic modes of oscillation, and biological and ecologicalsystems.

In a first class of implementations, square wave enveloping eventsoccurring between pairs of square wave signals may be detected byidentifying consecutive opposite transitions in one signal occurringbetween consecutive opposite transitions of the other signal, and viceversa.

In a second class of implementations, instantaneous values of two squarewaves may be collectively regarded as a symbol of asynchronous state.Square wave enveloping events are detected by identifying signaturesymmetries in the resulting sequence of symbols. Realizations of thissecond class of implementations amount to interpreting the relativevalues of the two applied square waves as a type of symbolic dynamics towhich pattern detection is applied. This class of implementations can beused to provide additional detailed information, such as a courseindication of relative phase, for example, utilizing specific symbolsequence signatures that can be detected in real time.

Although these two exemplary classes of implementations involvedifferent philosophies, they share many properties including the onesmentioned above. Either approach may be realized in software, hardware,and combinations thereof. If the provided square waves are notco-synchronized to an underlying clock, they may be sampled periodicallyfor one level of accuracy or performance, or implemented asynchronouslywith logic circuits and flip-flops at a higher level of accuracy orperformance. Sampling rate, race conditions, and transition detectionrecovery-time facilities that may be utilized in various implementationsdetermine the limits of maximal operational frequency range and minimummeasurable frequency difference. In hardware, for example, a typicallogic circuit implementation includes two to four flip-flops ortwo-stage two-bit shift registers and modest combinational logic. Theresulting system can thus be readily implemented as a utility integratedcircuit of modest “MSI” scale or as a small “IP core” withinlarger-scale system-on-a-chip realizations.

BRIEF DESCRIPTION OF THE DRAWINGS

The above and other aspects, features, and advantages of the presentinvention will become more apparent upon consideration of the followingdescription of preferred embodiments, taken in conjunction with theaccompanying drawing figures.

FIG. 1 a-1 f illustrate event-driven and time-driven symbol sequences aswell as conditions determined by state or by transitions between states.

FIGS. 2 a-2 b illustrate exemplary enveloping events between square wavesignals of differing frequencies.

FIG. 2 c formalizes aspects of FIGS. 2 a-2 b, illustrating eight typesof symmetry events, their constituent symbols, and associated notations.

FIGS. 3 a-3 b illustrate additional aspects and notations pertaining toeight types of symmetry events, their constituent symbols, andassociated notations.

FIG. 4 illustrates organizations of symmetry events with respect tocommon inner symbols and common outer symbols, showing these to beequivalent.

FIGS. 5 a-5 e illustrate various input signal and output signalarrangements pertaining to use of a symbolic processor. Additionally,FIGS. 5 d-5 e illustrate a more detailed view of the conversion oftime-varying signal waveforms to sequences of symbols and subsequentapplication of these to pattern detection systems or methods.

FIGS. 6 a and 6 d-6 g illustrate various exemplary clock-pulse creationmeans driven by input waveform transitions.

FIGS. 6 b-6 c illustrate input/output waveform relationships associatedwith FIGS. 6 a and 6 d-6 g.

FIG. 7 illustrates an exemplary approach to a symbol-based embodiment ofthe invention.

FIG. 8 a-8 d illustrate an exemplary approach to a symbol-basedembodiment of the invention. LED indicators are provided to indicatevarious operating conditions, states, and obtained results.

FIG. 9 illustrates a more concise symmetry-event detector structure thatmay be used in place of the general magnitude-comparator function shownin FIG. 8 b.

FIG. 10 illustrates an exemplary problematic implementation of aproblematic attempt at a transition-based embodiment of the invention,demonstrating the need to attend to details of interleaved transitionsbetween input waveforms as raised in conjunction with FIG. 2 b.

FIG. 11 a illustrates an exemplary successful implementation of onedetection step of a transition-based embodiment of the invention,demonstrating the ability to ignore the relative order of up and downtransitions of the lower-frequency waveform.

FIG. 11 b illustrates an exemplary exchange of input waveforms shown inFIG. 11 a so as to implement the complementary detection step of atransition-based embodiment of the invention.

FIGS. 11 c-11 e show exemplary transformational steps reorganizing thearrangement of FIG. 11 b so that it shares the same input structures asthat of FIG. 11 a.

FIG. 11 f shows an exemplary resulting superposition of the arrangementof FIG. 11 a with the arrangement of FIG. 11 e.

FIG. 12 a illustrates an logic-circuit implementation of the arrangementof FIG. 11 f.

FIGS. 12 b and 12 c illustrates the addition of LEDs to theimplementation of FIG. 12 a. The LEDs indicate various operatingconditions, states, and obtained results.

FIG. 13 illustrates an exemplary two-channel signal source for use indemonstrating, prototyping, and performing additional research anddevelopment of various aspects of the invention.

FIG. 14 a illustrates exemplary geometric localizations of various“IP-cores” in an exemplary “system-on-a-chip” (“SoIC”) implementation,wherein one or more of the IP cores may include an embodiment of theinvention.

FIG. 14 b illustrates a wider range of silicon-based embodiment optionsfor the invention.

FIG. 15 aillustrates an exemplary algorithmic embodiment of asymbol-based embodiment of the invention, while FIG. 15 b illustrates anexemplary algorithmic embodiment of a transition-based embodiment of theinvention.

FIGS. 16 a-16 c illustrate examples of how frequency comparatortechnology may be extended to handle more than two input signals, asprovided for by an embodiment of the invention.

FIG. 17 illustrates a descriptive representation of how acontinuous-time/continuous-state dynamical system may be collapsed intoa discrete-time/discrete-state dynamical system, relevant in the settingof symbolic dynamics (also known as topological dynamics).

FIGS. 18 a-18 c illustrate how the combined state-space trajectory of apair of continuous-time/continuous-state oscillators may be viewed as awrapping trajectory on the surface of a hollow torus.

FIGS. 19 a-19 d illustrate how different integer frequency ratiosbetween the two oscillators of FIGS. 18 a-18 c result in differingwrapping characteristics and trajectory slopes, employing a view of thetorus as joined edges of a flat tile.

FIG. 20 illustrates how the torus of FIGS. 18 a-18 c and 19 a-19 b maybe symmetrically quantized into regions associated with the symbolsemployed by the invention. In particular, the symmetric quantizationcorresponding to the case where the input signals are symmetric squarewaves is depicted.

FIG. 21 a illustrates a sequential tiling representation of thesymmetrically quantized torus of FIG. 20, wherein motion in the verticaldirection represents time and/or phase of one oscillator and motion inthe horizontal direction represents time and/or phase of the otheroscillator. FIG. 21 b illustrates an exemplary trajectory on thesequential tiling representation of FIG. 21 a and the identification ofan exemplary symmetry event.

FIG. 22 illustrates how varying the slope of the trajectory on thesequential tiling, the slope determined by the ratio of the frequenciesof the two oscillators, to values above or below unity, results indiffering symmetry events each giving complementary indication as towhich oscillator has the higher frequency.

FIG. 23 illustrates exemplary portions of trajectories associated witheach of the eight symmetry events associated with an embodiment of theinvention. The four symmetry events on trajectories with slopes greaterthan unity are uniquely associated with one oscillator being faster,while the four symmetry events on trajectories with slopes less thanunity are uniquely associated with the other oscillator being faster.

FIG. 24 illustrates a larger scale view of a portion of a trajectory ofan exemplary pair of waveforms whose relative ratio of oscillatingfrequencies varies in some intervals in time.

FIGS. 25 a-25 d illustrate exemplary symbol generation phenomenaresulting from applying asymmetric pulse waveforms as input signals. Inparticular, such input waveform asymmetry, when applied toimplementations of the invention designed specifically for symmetricinput waveforms can create problematic alternating indications that eachof the oscillators is faster than the other when the two oscillatorfrequencies are sufficiently close together.

FIG. 26 illustrates adaptations to the exemplary symbol-based embodimentof FIGS. 8 a-8 c that provides for the handling of asymmetric pulsewaveforms.

FIG. 27 illustrates a variation of the quantized torus of FIGS. 19 a-19d, which is adapted to asymmetric pulse waveforms.

FIG. 28 a illustrates an asymmetric sequential tiling representation ofthe asymmetrically quantized torus of FIG. 27.

FIG. 28 b illustrates exemplary trajectories associated with fixedfrequency ratios of unity, less than unity, and greater than unity intheir traversal over the asymmetric sequential tiling.

FIG. 28 c illustrates exemplary portions of trajectories associated witheach of the eight symmetry events associated with the invention.

FIG. 29 a illustrates timing notations that may be applied to asymmetricinput waveforms.

FIG. 29 b illustrates the interval of evolving occultations ofasymmetric aspects of two exemplary asymmetric input waveforms withfrequencies sufficiently close together to result in alternatingindications that each of the oscillators is faster than the other.

FIGS. 30 a-30 d illustrate behavioral signatures of exemplaryalternating indications that each of the two exemplary oscillators isfaster than the other resulting from asymmetric input waveforms andfrequencies sufficiently close together.

FIGS. 31 a-31 c depict what occurs for waveforms with some asymmetry asthe frequency ratio is increased from a value sufficiently lower thanunity, through ratios sufficiently close to unity, and then to ratiossufficiently greater than unity.

FIGS. 32 a-32 c illustrate the evolution of symbol sequences andsymmetry events before, during, and after occultations of asymmetricaspects of two exemplary asymmetric input waveforms with frequenciessufficiently close together.

DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS

In the following detailed description, reference is made to theaccompanying drawing figures which form a part hereof, and which show byway of illustration specific embodiments of the invention. It is to beunderstood by those of ordinary skill in this technological field thatother embodiments may be utilized, and structural, electrical, as wellas procedural changes may be made without departing from the scope ofthe present invention.

Various embodiments of the present invention utilize “enveloping” eventphenomena, intrinsic to the dynamics of pairs of square waves ofdifferent frequencies, to determine which of the pair is of a higherfrequency than the other and potentially other information. In general,the enveloping events may be detected by monitoring the pattern ofstates or state transitions associated with the pairs of square waves.

State View of the Dynamics of a Square Wave Pair

At a high level, state may be associated with pairs of square waves bytreating the instantaneous measured value of the two square waves as atwo-component vector. For example, a first square wave signal A and asecond square wave signal B may each tale on values of 0 or 1 at anyparticular time (ignoring noise and transition-related transientphenomena). There would be four resulting states, as may be catalogedand named, for instance, by the symbols S₀, S₁, S_(2,) and S₃ as in theexemplary list of Table 1 set forth below: TABLE 1 S_(2A+B) = A B S₀ = 00 S₁ = 0 1 S₂ = 1 0 S₃ = 1 1This symbol assignment may be given by the following formula:S_((2a÷b))  (Eq. 1)where “a” is the instantaneous value of A (i.e., either {0,1 }) and “b”is the instantaneous value of B.

The first square wave signal A and the second square wave signal Btypically originate from an exogenous signal source and may be measuredin asynchronous (effectively) continuous time or insynchronously-sampled discrete time. Each type of measurement creates atemporal sequence of the symbols S₀, S₁, S₂, and S₃.

FIG. 1 a shows the case for continuous-time measurement, which producesan “event-driven” sequence of symbols, while FIG. 1 b shows the case forsynchronously-sampled discrete-time measurement, which produces a“time-driven” sequence of symbols. Referring to FIG. 1 a, the graphs ofa first square wave signal A 110 and a second square wave signal B 120,each of which is allowed to take on one of two values at any given time,are shown evolving in time, with time increasing from left to right.

In FIG. 1 a, first square wave signal A 110 is shown progressing throughan “up” transition 111 between a previous “lower” value and a subsequent“higher” value, followed later in time by a “down” transition 112between the subsequent “higher” value and further subsequent return tothe previous “lower” value. This is followed by additional subsequent“up” and “down” transitions. Similarly, the second square wave signal B120 is shown progressing through “down” transitions 121, 123, an “up”transition 122, as well as additional subsequent transitions. Betweeneach of the transitions 121, 111, 122, 112, 123, etc. the pair ofwaveforms maintain a fixed state corresponding to one of the symbols S₀,S₁, S_(2,) and S₃, and the state changes to another symbol after thenext transition. Thus, any transition (e.g., 121, 111, 122, 112, 123,etc.) of either of the two square wave signals A 110 or B 120 causes astate transition, or symbol transition, event 131, 132, 133, 134, 135,etc. between which the state is held constant.

For example, in FIG. 1 a, the state just prior to transition event 131is S₁ (A=0, B=1), the state between transition event 131 and transitionevent 132 is S₀(A=0, B=0), the state between transition event 132 andtransition event 133 is S₂(A=1, B=0), the state between transition 133and transition 134 is S₃(A=1, B=1), etc. The result is an “event-driven”sequence of symbols {S₁, S₀, S₂, S₃ . . . }.

FIG. 1 b shows the case for synchronously-sampled discrete-timemeasurement, which may produce a “time-driven” sequence of symbols. Herethe values of the same first square wave signal A 110 and the samesecond square wave signal B 120 are periodically measured at individualsample times, denoted by sample times 141-150, and the value of a statemeasured at one sample time is maintained until the next sample time.Such an arrangement is useful in regular clock-driven signal processingimplementations.

In the example of FIG. 1 b, the state at sample time 141 is S₀, thestate at sample time 142 is S₂, the state at sample time 143 is S₃, etc.Note that the state at the two consecutive sample times 147 and 148 isS₂. If the rate of sampling were considerably faster than that depicted,situations where the same state is held for consecutive sample timeswould frequently occur. By definition, the event-driven symbol sequencecannot have consecutively repeated symbols (since a driving “event”corresponds to a change in state, hence change in symbol). Thus, ingeneral, for the same pair of square waves, an “event-driven” sequenceof symbols will typically differ from the “time-driven” sequence ofsymbols. FIGS. 1 e-1 f provide a comparison of permissible statetransitions among the states represented by symbols {S₀, S₁, S_(2,) S₃ .. . } for event-driven and time-driven measurements.

Referring ahead to the event-driven case of FIG. 1 e. direct transitionsbetween symbol pairs S₀ and S₃ and between symbol pairs S₁ and S₂ areforbidden as either would require both square wave signals A 110 and B120 to change states simultaneously, a physically impossible conditionexcept in pathological cases and even then overruled by circuitry raceconditions, as is well known to those skilled in the art of electronicdigital circuit design. Also, in the event-driven case of FIG. 1 e, eachstate may transition only to another state, not back into itself; thisis because, by definition, if there is no observed state transitionthere is no new event. Hence no repeated event symbols are possible.Taken together, the forbidden state transitions are those where thecurrent symbol and the immediately previous symbol are either equal(transition back to same state) or complements of one another (bothsquare wave signals A 110 and B 120 change states simultaneously).

In the time-driven case of FIG. 1 f; transitions from a state back intoitself are not only possible but dominate the time-driven symbolsequence as the sampling rate increases. As is clear to one skilled inthe art, however, if the sampling rate is high enough to capture theeffect of every transition in each of the pair of square waves (i.e., asampling rate of at least twice the frequency of the highest-frequencysquare wave), the resulting time-driven event sequence can betransformed into an approximate event-driven sequence (such as that ofFIG. 1 a) where the only errors introduced are time-quantization delays.This transformation may be done for example, by omitting any repeatedsample values such as 148 in FIG. 1 b. Additionally in the time-drivencase of FIG. 1 f, direct transitions between symbol pairs S₀ and S₃ andbetween symbol pairs S₁ and S₂ are in some circumstances possible, forexample:

-   -   if the sampling rate is slow enough (the faster the sampling        rate, the less likely this situation will occur);    -   if the square wave signals A 110 and B 120 are digitally        generated, of frequencies that are ratios of integers, and        phase-locked.        These sampling-rate artifact transitions are indicated by the        dashed lines in FIG. 1 f. Care should be taken to adequately and        stably handle cases where the sample time effectively coincides        with a transition in one of the waveforms, as with sample time        149.

Whether obtained directly as in FIG. 1 a, or derived from a time-drivensymbol sequence, the measurements of the two square waves ultimatelyprovide an actual or approximate event-driven symbol sequence. Themeasurements themselves may be made on the sustained values of thesquare wave, as called out by the bolded portions 161, 162, 163, 164,165 of the square wave in FIG. 1 c, or may be made on the transitions ofthe square wave, as called out by the bolded arrows 170, 171, 172, 173,174, 175 of the square wave in FIG. 1 d. For the measurement ofsustained values of the square wave, a “low-pass” filter or system forthe detection of a repeated value across a plurality of consecutivesample times may be used. For the measurement of the transitions of thesquare wave, a “high-pass” filter, an edge detector (employingstructures such as that of FIGS. 6 a-6 g, to be discussed later), or asystem for the detection of a change in value between consecutive sampletimes, may be used.

Enveloping Event Phenomena

With these concepts in place, the “enveloping-event” phenomena peculiarto square waves of different frequencies will now be described. FIG. 2 ashows again a first square wave signal A 210, here in particular calledout as having a lower frequency and thus a longer, wider-spread periodcompared with that of a second square wave signal B 220. This figuredepicts two special events 231, 232 where the second square wave signalB 220 makes both an up transition 233 and a down transition 234 duringan interval 235 where the first square wave signal A 210 is unchanged,On either side of these transitions in the second square wave signal B220, the first square wave signal A 210 makes its up transition 237 anddown transition 236. In this sense, an up-down or down-up “pulse” of thesecond square wave signal B 220 is enveloped by an up-down or down-up“pulse” of the first square wave signal A 210, and this can clearly onlyoccur if the frequency of B is higher than the frequency of A.

If the frequency of B is sufficiently higher than that depicted in FIG.2 a, even more transitions of the second square wave signal 220 would beenveloped within an up-down or down-up “pulse” of the first square wavesignal 210. An example of this can be found in FIG. 2 b, where severaltransitions 254, 255, 256, 257 of the square wave signal A₂ 253 areenveloped by an up-down “pulse” 258 of the square wave signal 252. Thus,a sufficient condition for a first square wave to have a lower frequencythan a second square wave is for there to be at least one consecutivepair of “up” and “down” transitions of the second square wave between aconsecutive pair of “up” and “down” transitions of the first squarewave. This condition will be referred to as an “enveloping event.”

FIG. 2 a illustrates two exemplary enveloping events, namely enveloping1 231 and enveloping event 2 232. As an example, eight types ofenveloping events may occur, and these will be discussed in conjunctionwith FIG. 2 c, after the following remarks.

First, although an enveloping event is sufficient for one square wave tobe determined as having a higher or lower frequency than another, it isnot a necessary condition. For example, if two square waves of differentfrequencies are phase-locked, there may be many classes of conditionswhere enveloping events cannot occur. Similarly, if two square waves aresufficiently close in frequency (for example, originating from twocesium clocks), the two square waves are effectively phase-locked forthe probable application interval (or lifetime) of the system. However,in many applications the two square waves are from separate sources andconditions that are not phase-locked and at frequencies sufficientlydifferent so that enveloping events naturally and regularly occur.Further, in typical phase-locked applications, enveloping events can beselectively created or prevented using frequency-shift and phase-shiftmodulation. Such applications are generally used in communicationssystems.

Second, the enveloping events can be detectable over a wide range offrequencies. The limiting case is where the frequencies of two squarewaves are very close. Referring to FIG. 2 b, if the frequencies of thetwo square waves A₁ 251 and 252 are very close, the detectionarrangement must be able to resolve narrow widths 250 a, 250 b of theenveloping of square wave B 252 by square wave A₁ 251. Also, morefrequently than not widths 250 a, 250 b of enveloping will be asymmetricand at times considerably so, thus requiring even higher performance inresolving narrow-widths of enveloping 250 a, 250 b.

Third, for full range of operation, the arrangement for detectingenveloping events must strictly determine that both square waves haveseparately completed their consecutive pair of “up” and “down”transitions, Simply detecting that a first square wave has had aconsecutive pair of “up” and “down” transitions with the value of asecond square wave having the same value, as might be attempted insimple implementations involving edge-triggered D flip flops, willprovide one or more false results. As an example of such a false result,note that both square wave A₁ 251 and A₂ 253 have the same value on thesquare wave B 252 up transition 259 as they do on the subsequent squarewave B 252 down transition 260; however, clearly the frequency of squarewave A₁ 251 is less than the frequency of square wave B 252 while squarewave A₂ 253 has frequency greater than the frequency of square wave B252.

Fourth, it is noted that the approach described thus far can besensitive to square wave asymmetry. At least some enveloping eventconditions can be violated if pulse widths are not 50%, and envelopingevents can be falsely generated if duty cycles are extreme with respectto the difference in periods of the two waveforms. This is a conditionthat is endemic to frequency measurements where square waves are needed,and a common solution is to preprocess each original square wave with atoggle flip-flop frequency divider as described in, for example, thework entitled CMOS Cookbook, Second Edition, by Don Lancaster, revisedby Howard Berlin, published by Newnes and Howard Sams, Inc., Boston, pp.276-277 (1988).

Finally, note that if a first square wave has a lower frequency than asecond square wave, the transitions of the second square wave typicallyhappen at a faster rate than the transitions of the first square wave.In subsequent discussions this is a useful dominating concept, so theterminology “X faster than Y” will be useful as a name for the conditionwhere the frequency of a square wave X is higher than the frequency of asquare wave Y.

Enveloping Events as Symmetry Events in Consecutive States and Some ofTheir Properties

FIG. 2 c depicts eight types of enveloping events. Enveloping may bewith either of the square waves having a given or opposite polarity,giving four types of events. Either square wave may be the “faster”(higher frequency) wave, giving two cases for these four types, or eightcases altogether.

Of value in both cataloging these and in subsequent analysis, it isuseful to characterize the cases using the state symbols S₀, S₁, S₂, andS₃ introduced earlier. The result in so doing is the following:

Cases where B 270 a is faster than A 270 b:

-   -   S₂S₃S₂ 275 a    -   S₃S₂S₃ 275 b    -   S₀S₁S₀ 275 d    -   S₁S₀S₁ 275 d

Cases where A 270 a is faster than B 270 b:

-   -   S₁S₃S₁ 276 a    -   S₃S₁S₃ 276 b    -   S₀S₂S₀ 276 c    -   S₂S₀S₂ 276 d        Thus the “signature” of one square wave having a faster rate        (higher frequency) than another are “symmetry events” of the        form:        S_(p)S_(q)S_(p)   (Eq. 2)        as indeed it is simply impossible for two square waves of the        same frequency to have these symmetric symbol sequences.

FIG. 3 a summarizes the findings of FIG. 2 c in a state-oriented form310 a-310 d and 330 a-330 d, rearranging the ordering to index the outersymbols in ascending order. The state is also indicated in vector form320 a-320 d and 340 a-340 d, representing separate samples, 2-bit wordsin a shift register, etc. It is equally viable to represent the findingsof FIG. 2 c in a transition-oriented form, indicating which square wavehas enveloping transitions, in which order these transitions occur, andwhat value the other square wave maintains throughout the envelopingevent. Further, it is useful to more concisely name each of the eightsymmetry events with a “symmetry event symbol”; for example, using thenotation:w_(pq)=S_(p)S_(q)S_(p)   (Eq. 3)

FIG. 3 b consolidates this notation, the results of FIG. 3 a, and arepresentation of transition-oriented forms into a single table. As anexample, this representation of transition-oriented forms may berendered according to the following rules:

The faster square wave is represented with a pair ofdirectionally-explicit transition arrows (e.g. 360 a-360 b) reflectingtheir consecutive order of occurrence in the enveloping event.

The slower square wave is represented with a horizontal line. Thishorizontal line is drawn above (e.g. 365 a) the pair of transitionarrows if the slower square wave maintains a high value throughout thetransition event, and is drawn below (e.g. 365 b) the pair of transitionarrows if the slower square wave maintains a low value throughout thetransition event.

The named source of each square wave (i.e., A or B) is written to theleft of its representation (e.g. 370 a-370 b).

FIG. 3 b shows there are a number of striking structural relationshipsexhibited, suggestive of possible underlying permutation group phenomenaand worthy of further study. Some of the structural relationships may beof value in various implementations, allowing in some situations useful“don't care” simplifications in combinational logic Karnough maps,algorithm design, and the like, Independent of implementation, however,FIG. 4 draws attention to two particular views of exhibited structuralrelationships and reveals yet more, perhaps unexpected, inherentstructure. The left table 401 is organized with common center symbol 405indexed in increasing order and listing the rate-distinguishing outersymbols 410, 415 which indicate which of the two square waves is faster.The right table 402 is organized exactly oppositely with the outersymbols 420 indexed in increasing order and listing therate-distinguishing common center symbol 425, 430 which indicate whichof the two square waves is faster. In fact, the two tables have exactlythe same entries. Further, the two rate-distinguishing columns 410, 415and 425, 430 in both tables are, reading from top to bottom, inretrograde (i.e., of opposite order).

Some further structural analysis will also be useful. First of all, andby way of non-limiting example, exclusive pairings are noted in theformation of symmetry events:

For B faster than A:

-   -   S₀ is always paired with S₁    -   S₂ is always paired With S₃

For A faster than B:

-   -   S₀ is always paired with S₂    -   S₁ is always paired witl S₃.        Next, in each of the frequency comparison cases “B faster than        A” and “A faster than B,” each of the symmetry events has a        unique “complement” (i.e., all 0's and 1's exchanged) within the        same frequency comparison case:

For B faster than A:

-   -   W₀₁=W₃₂*    -   W₁₀=W₂₃*    -   W₂₃=W₁₀*    -   W₃₂=W₀₁*

For A faster than B:

-   -   W₀₂=W₃₁*    -   W₁₃=W₂₀*    -   W₂₀=W₁₃*    -   W₃₁=W₀₂*        This is due to the fact that enveloping events occur with either        polarity. Note in all cases that:        W_(pq)=W_((3−p)(3−q))*   (Eq. 4)        in part due to the way the symbols {S₀, S₁, S₂, S₃ . . . } have        been indexed by the following formula:        S_((2a+b)).   (Eq. 5)

In a state-oriented implementation, the sequence of measured symbols maybe examined for the occurrence of eight possible symmetry event symbolsso as to determine which square wave signal is faster (i.e., has thehigher frequency). For an event-driven sequence (i.e., one precludingimmediate symbol transitions back into themselves) a symmetry event maybe detected by comparing the current symbol value to the symbol valuethat is two events in the past: if they are identical, a symmetry eventhas just occurred. Once a symmetry event has been detected, it may beclassified as a particular one of eight possible symmetry event symbolsbased on the values of the current symbol and immediately precedingsymbol, following from the definitions of the symmetry event symbolsw_(pq) shown in the following Table 2: TABLE 2 S_(current) S_(previous)Symmetry- Frequency A B A B Event symbol Relationship 0 0 0 0 0 0 0 1w₀₁ B faster than A 0 0 1 0 w₀₂ A faster than B 0 0 1 1 0 1 0 0 w₁₀ Bfaster than A 0 1 0 1 0 1 1 0 0 1 1 1 w₁₃ A faster than B 1 0 0 0 w₂₀ Afaster than B 1 0 0 1 1 0 1 0 1 0 1 1 w₂₃ B faster than A 1 1 0 0 1 1 01 w₃₁ A faster than B 1 1 1 0 w₃₂ B faster than A 1 1 1 1

Reorganization of columns (by partitioning each symbol into its A and Bcomponents and arranging like components in adjacent columns) yields aperiodic clustering such as that presented in Table 3 below: TABLE 3S_(current) S_(previous) S_(current) S_(previous) Symmetry- Frequency AA B B Event symbol Relationship 0 0 0 0 0 0 0 1 w₀₁ B faster than A 0 01 0 w₁₀ B faster than A 0 0 1 1 0 1 0 0 w₀₂ A faster than B 0 1 0 1 0 11 0 0 1 1 I w₁₃ A faster than B 1 0 0 0 w₂₀ A faster than B 1 0 0 1 1 01 0 1 0 1 1 w₃₁ A faster than B 1 1 0 0 1 1 0 1 w₂₃ B faster than A 1 11 0 w₃₂ B faster than A 1 1 1 1

The approach will be used directly to construct an exemplarystate-oriented implementation of a frequency comparator and, after laterdiscussion, detection of waveform asymmetries. In the two tables above,note that the remaining eight of the sixteen possible conditions are notrecognized as a symmetry event. Each of these unrecognized casescorrespond to the previously described forbidden state transitions wherethe current symbol and the immediately previous symbol are either equal(transition back to the same state) or complements of one another (bothsquare wave signals A and B change state simultaneously).

Both interesting and useful additional theory can be developed for theseand further structural observations, for example,

-   -   further algebraic relationships,    -   topological and geometric representations,    -   connections to formal established symbolic dynamics theory, and    -   other phenomena with unique sequence phrase “signatures.”

The discussion will return to these observations for further aspects ofthe invention. For the present, discussion is next directed towardapplying the analysis developed thus far to exemplary implementationsand applications of frequency comparison.

Exemplary Implementations

Two implementations will now be considered. First, exemplarysymbol-based approaches are presented, and then exemplarytransition-based approaches will be presented. These implementationsform a foundation readily extensible to implementing additional aspectsof the invention involving the aforementioned additional theory andfurther structural observations. Before beginning, attention is directedto the acceptance and handling of various types of input signals.

A general setting for signals directed to general implementations of theinvention is illustrated in FIG. 5 a. Here two sources of binary-valuedrectangular waveform signals 510 a-510 b (i.e., binary-valued symmetricsquare waves, or pulse waveforms with a duty cycle other than 50%) arepresented as input signals to a state machine or other electrical,algorithmic, computational, optical, mechanical, chemical, biological,or ecological system 520 configured to operate as a symbolic processor.These input signals may, for an applicable duration of time, befixed-periodic signals, time-modulated signals, frequency-modulatedsignals, and the like. The symbolic processor is shown producing one ormore output signals.

In many situations input signals presented to the symbolic processor maycomprise waveform types other than binary-valued rectangular waveformsignals. FIG. 5 b-1 illustrates one approach for handling this situationwherein these alternate types of input waveforms 530 a, 530 b are firstprovided to preprocessing operations 550 a, 550 b that convert thesealternate types of waveforms 530 a, 530 b into the binary-valuedrectangular waveform signals assumed in FIG. 5 a. These pre-processingoperations 550 a, 550 b may, for example, comprise one or more of thefollowing:

-   -   level-quantizing or comparator operations,    -   symbol recognition or conversion,    -   event recognition,    -   multiple-input signal aggregation, and    -   intra-media signal transduction.

In other implementations, input signals 530 a, 530 b comprising types ofwaveforms other than the binary-valued rectangular waveforms may be suchthat the waveformns themselves possess other types of symbolicattributes intrinsically recognized by a corresponding specializedimplementation of a symbolic processor. FIG. 5 b-2 depicts such anembodiment wherein input signals 530 a-530 b are applied directly to acorresponding type of symbolic processor 540.

In many situations where the input signals are indeed binary-valuedpulse waveforms, these waveforms may not be 50% duty cycle square waves.In fact in most real-life systems and situations, what appear to becompletely symmetric square waves reveal upon close inspection a dutycycle slightly different than 50%. This may be the result of slightsystem instabilities, non-ideal system characteristics, slight systemoperational errors, etc. In other situations there may be deliberatevariations in signal pulse width, As will be discussed later, dutycycles other than 50% can introduce additional effects and more complexbehavior, requiring more careful treatment as pulse width deviates forwaveforms of frequencies sufficiently close together. However, inputbinary-valued rectangular waveforms with duty cycles other than 50%signals may be successfully applied to a symbolic processor 520 designedfor more precisely-symmetric square waves by preprocessing both signalsby edge-triggered (toggle flip-flop) frequency dividers. This techniquecan be used to create symmetric square waves from only the rising oronly the falling edge of the original binary-valued input signalwaveforms. The applied signals are lower in frequency but retain manykey properties, in particular the condition as to which input signalfrequency is higher. This arrangement is depicted in FIG. 5 c with inputsignals 560 a, 560 b. Additionally, this technique may be applied anynumber of times to reduce ultra-fast original signals (as may arise frommeasurements).

Symbol-Based Embodiments

FIG. 5 d illustrates a high-level view of an exemplary class ofsymbol-based embodiments. Two square waves 570 a, 570 b are presented toa system or method 501 for interpretation as (or transformation to) asequence of symbols 502. This sequence of symbols 502 may be presentedto a pattern detection system or method 503 to produce one or moreoutput signals, flags, or conditions 590. This class of symbol-basedapproaches is a special case of the arrangement of FIG. 5 e in which aplurality of signals 580 a-580 n, each at a given moment talking one ofa plurality of values, and which may or may not be relatively periodic,are applied to a system or method 511 for interpretation as (ortransformation to) a sequence of symbols 512 which are presented to apattern detection system or method 513 to produce one or more outputsignals, flags, or conditions 595.

Referring again to FIG. 5 d, the system or method 501 may be anasynchronous logic circuit, a synchronous sampling system, a precedingalgorithm, etc. In the case where the system or method 501 isimplemented as a synchronous sampling system, transformations to anevent-driven sequence, such as those described earlier, may be employedto create sequence of symbols 502. Typically, such a realization willautomatically provide delineation between individual symbols withinsequence 502. If it does not, or in the case of an asynchronous logiccircuit or other asynchronous environment, such delineation betweenindividual symbols must generally be synthesized or derived.

There are a number of ways in which clock signals can be derived fromtransitions of a given square wave signal. The simplest of theseinvolves capacitive-coupling, as is well known to those skilled in theart; an example of this will be employed later in the circuit depictedin FIGS. 12 a-12 b, as will be discussed in more detail below.

A capacitive-coupling approach has frequency-range limitations, so FIGS.6 a-6 g illustrate additional ways in which clock signals can be derivedfrom transitions of a given square wave signal. FIG. 6 a shows aconfiguration comprising exclusive-OR gate 610 and time delay element620. When the input to this circuit experiences a logical valuetransition, the inputs to the exclusive-OR gate are briefly of differentlogical values. As shown in FIGS. 6 b and 6 c, this produces a pulse630, 640 comprising a width in time nearly that of the delay element 620of FIG. 6 a. This approach may be implemented in electronics, or withinan algorithm utilizing, for example, a delay operation and conditionaltest within a running loop.

FIG. 6 d shows the time delay realized by an analog RC circuit 650. Thepulse width created here is determined by the RC-time constant and thelogic threshold of the exclusive-OR gate. FIG. 6 e shows the time delayrealized by a pair of inverters 660 a, 660 b. The pulse width createdhere is approximately two gate propagation times. FIG. 6 f shows thetime delay realized by a single positive-logic gate 670. An AND gate isshown in FIG. 6 f, but other types of gates may be used. The inputs areshown jointly connected, but as appropriate for the type of logic gateused one input may alternatively be tied high or low. The pulse widthcreated here is approximately a single gate propagation time.

FIG. 6 g illustrates an exemplary way of generating complementary pulseswith essentially identical wavefronts and durations to minimize raceconditions. Here, the delay implementation of FIG. 6 e is used. The FIG.6 g implementation includes inverters 660 a, 660 b, but this arrangementcan readily be replaced by other delay implementations, such as thosedepicted in FIGS. 6 a, 6 d, 6 f, and the like. The transition pulseproduced is simultaneously applied to two symmetric-implementationExclusive OR gates 615 a, 615 b. A first of these two Exclusive OR gates615 b has its second input tied high, producing a logically identicaltransition pulse delayed by the Exclusive OR gate propagation time,while the second input of the other Exclusive OR gate 615 a is tied low,producing a logically-inverted transition pulse also delayed by the(typically nearly identical) Exclusive OR gate propagation time. Theresulting pair of complementary pulses, with nearly identical wavefrontsand durations, is of importance in some approaches to symbol transitiondetection implementation.

The acceptance and handling of various types of input signals thusaddressed, attention is now directed to exemplary symbol-basedembodiments.

FIG. 7 depicts an exemplary symbol-based embodiment based upon theexample of FIG. 5 d. Each of the two square wave input signals (A 710 aand B 710 b) is provided with a dedicated transition detector circuit720 a, 720 b (e.g., FIG. 6 a), and the resulting transition detectionpulses are combined (here by a subsequent OR gate 730) to create a“new-symbol-event” clock pulse. This clock pulse is used to clock a2-bit-wide shift register 740, to which the square waves are applied.Note that the 2-bit-wide shift register 740 driven by the combinedtransition detection pulses is used rather than two separately clocked1-bit shift registers. This is in accordance with the third remark inthe previous noted remark list.

The delay used in the two dedicated transition detector circuits 720 a,720 b is sufficient for the shift register to adequately perform shiftoperations. The combination of these delays and the combining logic gate730 create typically more than a two gate propagation time delay betweenthe arrival of a square wave transition at the shift register input andthe subsequent arrival of the clock pulse. This allows for a cleanclocked capture operation at the shift register inputs 750 a, 750 b. Theresult is an event-driven symbol sequence whose most recent threesymbols in the sequence are available for subsequent pattern detection.The instantaneous square waves, or their equivalents, together withtheir values at one and two clock pulses in the past, are presented to apattern detection circuit 760. The pattern detection circuit may includeat least combinational logic (and perhaps state-retained logiccomprising elements such as flip-flops, additional shift registers,etc.), resulting in one or more outputs 770 a-770 n derived from patterndetection operations.

FIGS. 8 a-8 d show an exemplary demonstration circuit based on theprinciples described so far, and may be constructed from standardlow-level logic TTL and CMOS chip families. Additional designtransformations and considerations have been included regarding theopportune use of spare gates available in multiple-gate chip packages,adequate clock time needed for operation of the co-clocked pair ofshift-registers, and the like. Alternatively, ASIC/PAL cells may beemployed.

In FIG. 8 a, each input 801 a-801 b is buffered using buffers 805 a and805 b, respectively, to produce a well-defined internal signal which isthen directed to transition detectors 810 a and 810 b, respectively, ofthe style depicted in FIG. 6 a. This arrangement is similar to that ofFIG. 6 f, but instead utilizes spare Exclusive-OR gates 815 a and 815 b,respectively, as delay elements 620 (FIG. 6 a) rather than the AND gate670 (FIG. 6 f).

The transition detections are directed to a logic OR operation 820 whichafter inversion 825 is suitable to clock a pair of 74195-series (TTL orCMOS) shift registers 830 a, 830 b. One shift register 830 a isconfigured to store the past two values (with respect to symbol eventchanges) of input A 801 a, while the other shift register 830 b isconfigured to store the past two values (with respect to symbol eventchanges) of input B 801 b. The current values of inputs A 801 a and B801 b, respectively notated A₀ 840 a and B₀ 840 b, as well as their mostrecent earlier value, respectively notated A⁻¹ 845 a and B⁻¹ 845 b, aswell as their next previous earlier value, respectively notated A⁻² 850a and B⁻² 850 b, provide signals applicable to determining the presenceand type of symmetry event that may be present at any given instant.

These six signals 840 a-840 b, 845 a-845 b, and 850 a-850 b are directedto the exemplary circuit of FIG. 8 b. Here, for the sake of simplicityin discrete logic chip realization, symmetry-events are detected by amagnitude comparator, and full-range primitive pattern detection isperformed by selected outputs of a de-multiplexer chip 855 (manyalternative arrangements are also possible as is clear to one skilled inthe at of basic logic circuit design). Also for the sake of simplicityin discrete logic chip realization, otherwise needed logic gates havesubsequently been saved by employing the “a>b” and “b<a” outputs of themagnitude comparator “NOR-ed” together by the negative-logic enable pinsof the de-multiplexer 855 to equivalently perform the simple operationof enabling the de-multiplexer on detection of a symmetry event. Manyother arrangements are also possible, as is clear to one skilled in theart of basic logic circuit design.

Further as to FIG. 8 b, the sixteen outputs of the 4-bit 74154 series(TTL or CMOS) de-multiplexer identify the four “A faster than B” 865a-865 d and four “B faster than A” 860 a-860 d conditions. The remainingeight outputs corresponding to forbidden combinations are therefore notused here. Logic operations, such as OR-ing of the four “A faster thanB” conditions to create a single “A faster than B” output, OR-ing of thefour “B faster than A” conditions to create a single “B faster than A”output, and the like, can be performed. Of these, some of the eightviable symmetry event conditions can be omitted in trade-offs of circuitcomplexity and speed versus system performance requirements. Inappropriate contexts, the inversion of the symmetry event detectionsignal may be interpreted and used as indication of “No Symmetry Event”conditions (alternatively interpreted as “Ambiguity” conditions), as maylogical operations on derived “A faster than B” and “B faster than A”indications.

FIG. 8 c illustrates an adaptation of the circuit of FIG. 8 b featuringthe addition of a number of status-indication LEDs and symbol-indication7-segment displays to enhance the study and more explicitly demonstrateoperational principles of various embodiments disclosed herein. Thestatus-indication LEDs provided here include individual notice of theeight symmetry events {W₀₁, W₀₂, W₁₀,W₁₃, W₂₀, W₂₃, W₃₁, W₃₂}, as wellas indication of “No Symmetry Event” conditions (alternatively,“Ambiguity” conditions).

The various signals produced by the exemplary circuits of FIGS. 8 b or 8c, or their equivalents, may be further processed to obtain more generalor other derived information. For example, all four symmetry eventdetection output signals associated with the “A faster than B” condition{W₀₂, W₁₃, W₂₀, W₃₁} may be directed to a logical OR operation to createa general overall indication of “A faster than B,” and similarly allfour symmetry event detection output signals associated with the “Bfaster than A” condition{W₀₁, W₁₀, W₂₃, W₃₂} may be directed to alogical OR operation to create a general overall indication of “B fasterthan A.” As another example, SR latches or other storage methods may beused to retain results until a subsequent symmetry event detection.

As another example, logical operations may be performed on the threeindications of “Ambiguity,” “A faster than B” and “B faster than A”indications to derive an “A and B same frequency within resolution”indication. FIG. 8 d shows an exemplary circuit incorporating theseexamples and a few additional features. SR latches are used to retainthe last known outcome as to which frequency was faster. The circuitprovides a DPDT switch, selectively allowing the SR latches to be resetwhenever there is an “Ambiguity” (No Symmetry Event“) condition, orallowing the SR latches to ignore that situation and retain the lastvalue. Numerous other approaches and derived signals may be realized asis clear to one skilled in the art (and demonstrated later inconjunction with FIG. 26). One skilled in the art will recognize thatthe full functionality of a magnitude comparator is not needed to detectthe symmetry events. For example, the circuit of FIG. 9 could also beused.

Transition-Based Embodiments

Next, exemplary transition-based approaches for the invention areconsidered. These effectively set states of a plurality of latchingflip-flops responsive to the rising and falling edges of the two squarewaves, and apply combinational logic operations to the resulting statevalues.

As an orienting note, transition-based implementations effectively setthe state of a plurality of flip-flops with the rising and falling edgesof signals and invoke combinational logic operations. The implementationcircuits for the invention may at first appear at a high level tosomewhat resemble some types of edge-triggered frequency comparatorcircuits known in the art. However, these implementations are completelydifferent in principle, structure, and operation. Derived from thenovelties of the invention, the signal flow is entirely feed-forward(i.e., no stored-state feedback) and requires no quadrature signalinputs. These properties alone make the implementations to followentirely different from edge-triggered frequency comparator circuitsknown in the art.

A transition-based implementation may identify eight symmetry eventconditions separately, as was done in the state-oriented implementation,or in related groupings (“equivalence-classes”). In state-orientedimplementations, equivalence-classes are naturally implemented usingdon't care” conditions across the grouping of states (utilizing thecommon practice of Karnaugh maps). Such detailed states need not be keptin transition-based implementations; it is possible to realize “don'tcare” structures across time, i.e., detecting classes of grossly similarphenomena independent of the fine-structure in the temporal ordering ofevents. However, this must be done carefully to avoid the situationdepicted in FIG. 2 b, for example.

To illustrate this concern, FIG. 10 illustrates a problematic approach(one that ignores the third remark cited earlier). In this example, therising and falling transitions of each square wave are used to triggersampling of the value of the other square wave at that instant (1010a-1010 d). The resulting data appears at first readily useful, but failsto be definitive. Referring to FIG. 2 b, the approach of FIG. 10 willgive the same results when square wave B 252 is compared to relativelylower-frequency square wave A₁ 251 or relatively higher-frequency squarewave A₂ 253.

The key condition in the sequence of transitions that must be capturedis:

-   -   the slower square wave must malice a transition (either up or        down);    -   the faster square wave must make at least the next two        transitions (either up then down or down then up);    -   and only then may the slower square wave malice its transition.        Data capture and pattern detection thus operate in a manner not        unlike that of a combination lock that recognizes combination        codes. As an additional caution, this approach to a        transition-based implementation employs the use of both rising        and falling edges of the square waves. Thus the circuit involves        the co-presence of inversions and non-inversions of the same        signal. The generation and handling of these co-present        inversions and non-inversions of the same signal require care to        prevent unnecessary limitations due to race conditions.

FIG. 11 a illustrates a first step in an exemplary abstract logiccircuit realization of a transition-based implementation responsive toboth rising and falling edges of the symmetric square wave signals A1101 a and B 1101 b, and taking the above concerns into consideration.Here, four SR (“set-reset”) flip-flop latches 1103 a-1103 d are drivenby transients of inverted and non-inverted versions of binary waveformsA 1101 a and B 1101 b. The notation Q(S,R) denotes the state of thelatch output as a function of S and R: TABLE 4 S R Q(S, R) 0 0 Previousvalue of Q 1 0 1 0 1 0 1 1 1 (pseudo-stable)

Thus the configuration depicted in FIG. 11 a causes the four SR latchesto behave as follows:

-   -   The first latch output Q₁ is:        -   set to 1 when A makes a transition from 1 to 0;        -   set to 0 when B makes a transition from 1 to 0;    -   The second latch output Q₂ is:        -   set to 1 when A makes a transition from 0 to 1;        -   set to 0 when B makes a transition from 1 to 0;    -   The third latch output Q₃ is:        -   set to 1 when A makes a transition from 1 to 0;        -   set to 0 when B makes a transition from 0 to 1;    -   The fourth latch output Q₄ is:        -   set to 1 when A makes a transition from 0 to 1;        -   set to 0 when B makes a transition from 0 to 1,

The status of these latches 1103 a-1103 d can thus be used to keep trackof the relevant most recent sequential status of the rising and falling(up and down transitions) of each of the input waveforms. There are atleast two high-level approaches to employing subsequent combinationallogic to determine the occurrence of enveloping events relevant tofrequency comparison in accordance with embodiments of the invention:

-   -   Trapping the two conditions (a) and (b) where one waveform makes        both an up transition and a down transition during an interval        of time when the other waveform undergoes    -   (a) no up transition    -   (b) no down transition

and taking the logical “OR” of these (as either condition (a) or (b) canindicate which input waveform is at the higher frequency);

-   -   Trapping the conditions (c) and (d) where one input waveform        undergoes:    -   (c) an up transition    -   (d) an down transition

during an interval of time when the other waveform does not undergoeither, transition and taking the logical “AND” of these (as bothconditions (c) and (d) are required to determine which input waveform isat the higher frequency). The exemplary embodiment of FIGS. 11 a-11 f,described below, implements the latter approach.

If A 1101 a a makes no transition up nor down transition in an intervalwhere B 1101 b makes an up transition 1110 a, then outputs Q₁ 1105 a andQ₂ 1105 b are both set to 1, and remain in that condition until B 1101 aeither makes a down transition (which then causes Q₁ 1105 a to reset to0) or makes an up transition (which then causes Q₂ 1105 b to reset to0). Referring to cases 275 a-276 d of FIG. 2 c, these are two of thepartial conditions in which the symmetric square oscillation of B 270 bmust be faster than the symmetric square oscillation of A 270 a. An ANDoperation 1107 a acting on outputs Q₁ 1105 a and Q₂ 1105 b produces alogical 1 under these first two “B faster than A” cases.

Similarly, if A 1101 a makes no up nor down transition in an intervalwhere B 1101 b makes a down transition, then outputs Q₃ 1105 c and Q₄1105 d are both set to 1, and remain in that condition until A 1101 aeither makes a down transition (which then causes Q₃ 1105 c to reset to0) or makes an up transition (which then causes Q₄ 1105 d to reset to0). Referring to cases 275 a-275 d of FIG. 2 c, these are remaining twopartial conditions in which the symmetric square oscillation of B 270 bmust be faster than the symmetric square oscillation of A 270 a. An ANDoperation 1107 b acting on outputs Q₃ 1105 c and Q₄ 1105 d produces alogical 1 under these second two “B faster than A” cases.

A subsequent AND operation 1108 acting on previous AND outputs 1110 aand 1110 b, captures the condition where signal B makes both an up and adown transition during an interval with no transition of either type bysignal A. This situation corresponds to the four “A faster than B” cases275 a-275 d depicted in FIG. 2 c. When this condition is met, the ANDoperation 1108 ultimately produces a logical 1 if A 1101 a is fasterthan B 1101 b (at least where A 101 a and B 1101 b are symmetric squarewaves). Although not pursued here, it is noted that the three two-inputAND operations 1107 a, 1107 b, 1108 may be combined into a singlefour-input AND operation, which can be used to reduce chip count ifimplemented using standard small-scale logic ICs.

By replicating the arrangement of FIG. 11 a with the roles of A 1101 aand B 1101 b reversed, one obtains the arrangement of FIG. 11 b, whichultimately produces a logical 1 if A 1115 a is faster than B 1115 b,where A 1115 a and B 1115 b are symmetric square waves.

Even though the configurations of FIG. 11 a and 11 b use different latchdriving arrangements, they are very similar. In fact, either may beconverted to use the same latch driving arrangement as the other,allowing the two circuits to be readily merged into a form sharing thesame four SR latches. This will be illustrated by converting the latchdriving arrangement of FIG. 11 b into the latch driving arrangement ofFIG. 11 a. Because of the symmetry of the SR latch, one has theinversion relation:{overscore (Q)}(S,R)=Q(R,S)   (Eq. 6)Applying this to the values produced by the outputs of each of the fourSR latches 1117 a-1117 d in FIG. 11 b yields the equivalent latchoutputs 1135 a-1135 d depicted in FIG. 11 c. However, these complementedlatch outputs are in fact readily provided by an SR latch (as known tothose skilled in the art and later seen in FIG. 12 a) as the {overscore(Q)} output, thus immediately yielding the configuration of FIG. 11 d.

Reversing inputs A 1155 a and B 1155 b of FIG. 11 d now only causes theneed to swap outputs of the second 1157 b and third 1157 c latches,resulting in the configuration shown in FIG. 11 e. Comparing FIG. 11 ewith 11 a reveals the same latch driving arrangement, readily enablingthe combined circuit of FIG. 11 f.

FIG. 12 a shows a reference circuit realization of the arrangement ofFIG. 11 f, and utilizes standard TTL/CMOS series integrated circuits.The pulse transition edges of applied signals A 1200 a and B 1200 b areisolated by, for example, 0.01 uf capacitors 1205 a, 1207 a, 1205 b,1207 b driven by buffer and inverter circuits comprised of inverterelements 1209 a, 1211 a, 1213 a, 1209 b, 1211 b, 1213 b. Thistransition-driven pulse generation arrangement is a simplifiedalternative to the circuits of FIGS. 6 a-6 g, but has frequency-rangelimitations and can readily be replaced by circuits employing thetechniques depicted in FIGS. 6 a-6 g.

These four transition-driven pulse signals (W 1216 a, X 1216 b, Y 1216c, Z 1216 d) are applied to inverted-input RS-latches 1220 a-1220 dformed from pairs of NAND gates. The inverted property of the latchinputs, together with the choice of capacitively coupled signals, arealigned to match the latch-driving arrangements of FIG. 11 f.

The four (NAND gate, inverted-input) RS latches 1220 a-1220 d provideboth original and inverted versions of latch state outputsimultaneously, and these output signals are directed (inpositive-valued logic) to four additional NAND gates 1225 a-1225 d.These latter four NAND gates 1225 a-1225 d produce (in inverted-valuedlogic) signals corresponding to the Q₁ ^(A)&Q₂ ^(A) 1190 a, Q₃ ^(A)&Q₄^(A) 1190 b, Q₁ ^(B)&Q₃ ^(B) 1190 c, Q₂ ^(B)&Q₄ ^(B) 1190 d signals ofFIGS. 11 e and 11 f. The inverted values of these logic signals is inkeeping with inverted-value conventions of TTL/CMOS families of logicchips, should these signals be used for other purposes, and are alsoconvenient for direct driving of indicator LEDs. These inverted-valuelogic signals are then applied to two OR gates 1230 a-1230 b, which byDeMorgan's law act as NAND gates on positive-valued logic signals.

This gives inverted indications of “A faster than B” and “B faster thanA,” the inverted-value signals are in keeping with inverted-valueconventions of TTL/CMOS families of logic chips, and convenient fordirect driving of indicator LEDs. When actual square waves are applied,the indications of “A faster than B” and “B faster than A” areconstantly being reset when the slower waveform finally makes itstransition. Thus by applying the inverted “A faster than B” and “Bfaster than A” signals to another NAND gate inverted-input RS latch1235, the last indication of which of input signals A 1200 a and B 1200b had the higher frequency can be stored until that condition changes.

Further, when no condition indicating that one or the other of inputsignals A 1200 a and B 1200 b had the higher frequency is currentlyactive, there is no new conclusive information as to which of inputsignals A 1220 a and B 1220 b had the higher frequency. Under thesecircumstances, both of the outputs of the inverted “A faster than B” and“B faster than A” signals are high, so when they are applied to anadditional NAND gate 1240 a signal indicating “No New InformationAvailable” is produced, again in inverted-value form and thus in keepingwith inverted-value conventions of TTL/CMOS families of logic chips, andconvenient for direct driving of an indicator LED.

FIG. 12 b and 12 c show an LED demonstration adaptation of the referencecircuit realization of FIG. 12 a. Here a number of LEDs are added toindicate the various logic levels of the signal flow, demonstratingoperation. As indicated in the above discussion, the inverted-valuesignal conventions may be used for directly driving indicator LEDs tiedto the power supply and fitted with an appropriate current limitingresistor (for example 330 ohms for 5-volt TTL logic). Green LEDs, forexample, may be employed to represent states of affairs relating toinput signal A 1250 a being faster than input signal B 1250 b, and redLEDs may be employed to represent states of affairs relating to inputsignal B 1250 b being faster than input signal A 1250 a.

Using this convention, commonly available polarity-reversal bi-colorLEDs 1260 a-1260 d (green for one current direction, red for theopposite current direction) may be attached, as shown across thecomplementary-valued outputs of the four NAND gate inverted-input RSlatches 1255 a-1255 d; these, too, light as green for states of affairsrelating to input signal A 1250 a being faster than input signal B 1250b, and light as red for states of affairs relating to input signal B1250 b being faster than input signal A 1250 a. As shown in FIG. 12 b,individual single color LEDs 1265 a-1265 h may be added to redundantlyindicate the status of individual flip-flop outputs from the view pointof favorable conditions relating to the red and green convention. Thesignals produced by the four NAND gates of U2 are passed to FIG. 12 cwhere additional LEDs 1270 a-1270 d indicate the status of individualflip-flop outputs from the view point of favorable conditions relatingto the red and green convention. The indication of which signal isdeemed as being currently faster may be displayed with an additionalpair of LEDs 1275 a, 1275 b which also be color coded according to thecolor convention. As these light only at isolated moments, an additionalLED 1280, for example yellow, may be used to indicate “No NewInformation Available” status. The additional latch circuit may be usedto drive red and green LEDs 1290 a, 1290 b to indicate the last knownrelative speed determination, in which case the (yellow) LED 1280 may beinterpreted as a cautionary “No New Information Available” warningcondition to caveat the displayed latched relative speed indication.

The various LED indications introduced in FIGS. 12 b and 12 c readilyverify details of the design theory when inputs A 1250 a and B 1250 bare driven with signals from a (electromechanically-debounced)pushbutton. When driven with low-frequency symmetric square waveoscillators, the LED indications visually demonstrate the patternsresulting from the various relative square wave conditions that lead tothe frequency comparator output determination.

When driven with signals from (electromechanically-debounced)pushbuttons, the circuit plainly demonstrates another application: itprovides an indication as to which of two buttons or switches was lastcycled between on and off with no change in the status of the otherbutton or switch. This observation provides further motivation for achip implementation of at least the transition-based embodiment of theinvention as it can be used not only as a frequency comparator but alsoin user-interface, sequence-auditing, and other industrial applications.

Signal Source for Demonstration of Symbol-Based and Transition-BasedEmbodiments

FIG. 13 shows a demonstration signal source providing pushbuttonactuation and variable frequency low-frequency oscillators withcontrollable symmetry useful in exercising the configurations of FIG. 12b and FIGS. 8 a-8 c, for example. This exemplary demonstration signalsource 1300 comprises two nearly identical independent versions 1310 aand 1310 b of the same circuit; the only exception is switch 1320 thatallows for pulse widths to be set and adjusted separately (in which casethe two circuits 1310 a and 1310 b indeed are fully identical andindependent) or set with a common pulse width adjustment control, Eachnearly identical circuits 1310 a, 1310 b comprises an adjustablelow-frequency oscillator, internally comprising an integrator and acomparator in a positive feedback loop.

The comparator output is a binary-valued periodic waveform very closelyresembling (if not matching) that of a symmetric square wave, while theintegrator output is a continuous-valued periodic waveform very closelyresembling (if not matching) that of a symmetric triangle-wave. Such alow-frequency oscillator is well known in the art of analog musicsynthesizers. The comparator compares the integrator triangle waveoutput to a fixed reference voltage set to a value that is half that ofthe amplitude of the triangle wave.

Circuit 1300 of FIG. 13 provides this reference voltage by buffering theoutput of a trimpot that may be adjusted for maximal symmetry of theoverall output (or for precise duty-cycle matching when the two halves1310 a and 1310 b of circuit 1300 share the same pulse-width adjustmentcontrol, to be described next). The output of the comparator is variablyattenuated by a logarithmic potentiometer panel control arrangement toprovide variable frequency control. A small resistor (here 10 ohms)between the logarithmic potentiometer panel control and the bufferedreference voltage sets the lower limit of the frequency range possiblewith the other components and configuration involved.

The triangle wave is additionally directed to a second comparator thatcompares the integrator triangle wave output to a freely adjustablevoltage set by a linear potentiometer panel control. This allowsvariation in the asymmetry duty-cycle of the resultant pulse waveform,as is well known in the art of analog music synthesizers, motor control,and communications. Switch 1320 allows, as described earlier, pulsewidths of the two oscillators to be adjusted separately with individualpanel controls (“INDEPENDENT” 1322) or together (“GROUP” 1324), sharingthe pulse width adjustment panel control 1326 for oscillator B.

Switches 1330 a, 1330 b are provided for each oscillator to permitselection of a nearly-symmetric square wave or a controllably asymmetricpulse waveform. The results are individually passed to respective toggleflip-flops 1335 a, 1335 b that restore symmetry under all conditionswhile also dividing the frequency by a factor of 2. Additional switches1340 a, 1340 b are individually provided to each oscillator so as topermit selection of the undivided (“ORIGINAL”) waveform or the symmetricfrequency-divided (“DIVIDED”) version. The resulting signal selectionsare then passed to yet another pair of switches 1345 a, 1345 b, whichare provided to each channel so as to permit selection of the selectedoscillator signal or a binary signal responsive to the position of adebounced pushbutton.

The pushbuttons are debounced using RS latches 1350 a-1350 b as is wellknown in the art of digital user interface design. The resulting choicesis then passed to inverters 1355 a, 1355 b, and additional switches 1360a, 1360 b are provided for each channel, These additional switches 1360a, 1360 b are configured to select between the original (“NORMAL”) andinverted (“INVERT”) version of the respective signals created thus far.

At this point the signals are directed to one or more additionalbuffering stages for driving external circuitry. Here inverters are usedto drive LEDs 1365 a, 1365 b, provide separate outputs 1370 a, 1370 b,1375 a, 1375 b, 1380 a, 1380 b, 1385 a, 1385 b to the symbol-based andtransition-based embodiments (such as those of FIGS. 8 a-8 c and FIGS.12 a, 8 b) as well as test instruments such as oscilloscopes, phasemeters, and frequency meters.

Other implementations may add additional inverters to change the logicsense of the signal outputs, employ op amps for better current drive andstatic electricity immunity, and the like. As to phase and frequencymeasurement, it is noted that the quite inexpensive Extech model MN26DVM provides adequately precise phase and frequency measurements for usewith the demonstration circuits of FIGS. 8 a-8 c and FIG. 12 a.

The demonstration signal source 1330 just described facilitatesrelatively deep introductory study of the properties of pairs ofasynchronous square wave waveforms and the present invention asdescribed thus far, as well as exploring the behaviors of pairs ofasynchronous asymmetric binary waveforms, to be discussed in asubsequent section.

Single-Chip System and Sub-System Embodiments

Symbol-based or transition-based embodiments may be implemented as asubsystem of discrete components, as a devoted functional integratedcircuit, as a configuration of standard cells in a gate array, FPLA,ASIC, VLSI on a chip, or as an “IP core” within a System-on-a-Chip(SoIC).

The design of FIG. 8 a can be readily adapted to an integrated circuitimplementation via appropriate transformations and simplifications. Forexample, the magnitude comparator chip reduces the wiring and chip countin the FIG. 8 a implementation, but has far more functionality than isrequired to detect symmetry-events; it may be replaced by a more focusedsymmetry-event detector such as that illustrated in FIG. 9. Similarly,the 4-bit de-multiplexer, convenient for showcasing the individualsymmetry-event detections, can be replaced with more focusedcombinational logic, potentially leveraging as opportune the extensivestructures and symmetries identified earlier. Additionally,shift-registers 830 a, 830 b may be restructured as alternativeflip-flop configurations. Further, the entire functionality, orvariations thereof, may be adapted into an algorithm, as will bedescribed later in conjunction with FIGS. 15 a-15 b.

Similarly, the arrangement of FIG. 11 f and/or its reference circuitrealization of FIG. 12 a may also be readily adapted to an integratedcircuit implementation. As described earlier, a chip implementation ofat least the transition-based embodiment of the invention can be usednot only as a frequency comparator but also in a user-interface,sequence-auditing, and other industrial applications.

FIG. 14 a illustrates a wide range of silicon-based options inaccordance with another embodiment of the invention. A chosen abstractoperating principle 1410 provided for by the invention may be realizeddirectly employing standard TTL/CMOS components 1420 in ways similar tothat described in conjunction with FIGS. 8 a-8 c, and FIGS. 12 a, 12 b.Alternatively, abstract operating principle 1410 may be realizedutilizing gate and/or cell primitives inherent to specific ASIC, FPLA,and related technologies 1430. Additionally, abstract operatingprinciple 1410 may be realized utilizing primitives inherent to aspecific fabrication process 1440.

In a more measured approach, abstract operating principle 1410 may beencoded into a simulation-oriented realization 1450. In some simulationenvironments, the simulation description may be directly applied tospecific ASIC, FPLA, and related technologies 1430. Other types ofsimulation environments interwork with Very-High-level DescriptionLanguage (VHDL) systems 1460 that in turn can be directly applied tospecific ASIC, FPLA, and related technologies or a specific fabricationprocess 1440.

FIG. 14 a also illustrates a number of ways the aforementionedrealizations may be adapted to create an IP core for use as a subsystemin a larger functional integrated circuit, gate array, FPLA, ASIC, VLSI,or SoIC, VHDL environments may be packaged to create a VHDL-defined IPcore 1470. Alternatively, fabrication process-specific realizations 1440may be packaged at a high-level to create a process-specified IP core1480, or packaged as a subsystem mask 1490 to create a mask-specified IPcore 1495.

FIG. 14 b illustrates an exemplary chip (integrated circuit) physicallayout that comprises an exemplary number of “IP core” subsystems 1401a-1401 f, all co-integrated to form a larger system in a largefunctional integrated circuit 1400. This large functional integratedcircuit 1400 may be realized employing underlying gate array, FPLA,ASIC, VLSI, or SoIC technology. It is in this context wherein anembodiment of the present invention may comprise or be comprised by oneof the subsystems 1401 a-1401 f.

Algorthmic Embodiments

The invention further provides for an algorithmic implementation of thesymbol-based and transition-based embodiments presented earlier. As anexample, FIG. 15 a is a flowchart depicting an exemplary algorithmicembodiment of a symbol-based implementation operating in a time-drivenfashion as depicted in FIG. 1 f.

The ongoing dataflow begins with comparison 1502 of the current valuesof applied input signals 1500 a, 1500 b. If these input signals can besuccessfully and unambiguously quantized 1504 into well-defined binaryvalues 1508, the resultant pair of binary values 1508 (one for eachinput signal) are then converted 1510 to an appropriate “current symbol”{S_(O),S₁,S₂,S₃} 1512 as described earlier and summarized in FIG. 2 c.

This current symbol 1512 is compared 1514 to the previously stored valueof the “previous symbol” 1513 to determine if current symbol 1512 haschanged value since it was last stored. If it has not, no action isneeded 1516 (although current symbol 1512 may overwrite the identicalstored value of the previous symbol 1513 without consequence should thatprove useful in an implementation).

If the symbol has changed 1518, the value of previous symbol 1513 isretained 1522 as the new value of a “legacy symbol” 1526, and the valueof the current symbol 1512 is retained as the new value of the previoussymbol 1513. The current symbol 1512 and the legacy symbol 1526 arecompared 1530, and if they are identical, a symmetry event has occurred1534. The particular symmetry event is determined 1538 by the value ofthe current symbol 1512 and the value of the previous symbol 1513; thesemay be used either to directly determine which applied signal has thehigher frequency, or to compute 1538 a corresponding symmetry event“word” {W₀₁, W₀₂, W₁₀, W₁₃, W₂₀, W₂₃, W₃₁, W₃₂} 1540 which issubsequently interpreted 1542 to determine which applied signal has thehigher frequency 1544 a, 1544 b.

An exemplary algorithmic embodiment of a transition-based implementationoperating in a time-driven fashion as depicted in FIG. 1 f followsdirectly from the operational sequence depicted in FIG. 11 f and FIGS.12 a, 12 b. This would be understood by one skilled in the art, but anexemplary algorithmic flow chart is provided in FIG. 15 b. Referring tothe figure, applied binary-valued input signals A 1550 a and B 1550 bare checked for transitions 1552 a, 1552 b, and any detected transitions1554 a, 1554 b and 1556 a, 1556 b are used to change the binary state offour stored-value state variables Q₁, Q₂, Q₃, and Q₄ 1558 in a fashionconsistent with that of FIG. 11 f and FIG. 12.

These state variables are acted on by four AND operations 1565 a-1565 band 1568 a-1568 b, collectively producing results 1570 a, 1570 b, and1573 a, 1573 b corresponding to the Q₁ ^(A)&Q₂ ^(A), Q₃ ^(A)&Q₄ ^(A), Q₁^(B)& Q₃ ^(B), Q₂ ^(B), &Q₄ ^(B) signals of FIG. 11 f, and denoted as1190 a-1190 d. These results are then applied to two OR gates 1576 a,1576 b, producing pulsed indications of “A faster than B” and “EB fasterthan A” conditions 1580 a, 1580 b. These pulsed indications 1580 a, 1580b are then applied 1582 to set the state of a stored-value statevariable 1585 whose state denotes the last indication of which inputsignal A 1550 a or B 1550 b had the higher frequency. The pulsedindications are also directed to an additional AND operation 1587,producing a result 1590 indicating that “No New Information Available.”

It is understood that a wide range of alternative realization techniquesmay be used to implement various aspects of the invention as describedthus far, as well as additional aspects to be described in subsequentsections.

Extension to Three or More Input Signals

Provisions for handling three or more input waveforms are also possibleand provided for by the invention. This may be done in a number of ways.In one approach, the input waveforms are compared for all possiblepairs. The outcomes are then directed to additional logic operations orcomputations to identify which waveform is fastest, and potentiallyadditional ordering information. For example, consider the case of fourinput signals A,B,C,D. Four signals compared two at a time in allcombinations requires (by standard combinatorial formulas) sixcomparisons. If the pairwise comparison outcomes are, for example, asfollows:

-   -   (Frequency of C)>(Frequency of B)    -   (Frequency of C)>(Frequency of A)    -   (Frequency of C)>(Frequency of D)    -   (Frequency of B)>(Frequency of A)    -   (Frequency of B)>(Frequency of D)    -   (Frequency of A)>(Frequency of D)        then the overall uniquely determined result is:        (Frequency of C)>(Frequency of B)>(Frequency of A)>(Frequency of        D).   (Eq, 7)

In this first (“all pairwise combinations”) approach, for N inputsignals, the number of required frequency comparator stages grows asN!/[(N−2)!2!] with increasing number of inputs N. In a second(“cascading”) approach, the N input signals are separated into N/2groups. If N is even, the N/2 groups will comprise N/2 pairs. If N isodd, the N/2 groups will comprise (N−1)/2 pairs plus one additionalsignal. Each of the input signal pairs is independently compared tochoose the faster of each pair. The faster of each pair is passed to asubsequent stage where this process is repeated. A first stage will thuscomprise on the order of N/2 frequency comparators, a second stage willcomprise on the order of (N/2)/2N/4 frequency comparators, a third stagewill comprise on the order of (N/4)/2=N/8 frequency comparators, and soon. In this approach, for N input signals, the number of requiredfrequency comparator stages grows roughly as Nlog₂N. For N>6, the secondapproach is of less complexity, particularly if N is reasonably large.For N=20 the second approach has about half the complexity of the firstapproach, for N=40 the improvement is by a factor of approximatelythree, and for N=100 the improvement is by a factor greater than seven.

FIG. 16 a illustrates an exemplary circuit that propagates selectedinput 1603 from a pair of applied inputs 1600 a, 1600 b based on theoutcome of a frequency comparator. When the pair of applied inputs 1600a, 1600 b is the same as applied to the frequency comparator, thecircuit will propagate the input signal of faster frequency (or slowerfrequency, depending upon the connections and logic value assignments).

FIG. 16 b shows this arrangement being used with three frequencycomparators 1615 a-1615 c to identify the fastest of four input signals1610 a-1610 d. The results shown provide the faster of each pair ofcomparisons 1620 a, 1620 b and a logical indication 1625 a-1625 c as towhich of these is faster; these may be used directly if valuable in thatform or be directed to a subsequent instance of the circuit of FIG. 16 ato deliver the fastest (or in a complementary implementation, slowest)input signal. Alternatively, the logical indication as to which input ofeach pair is faster may be subjected to logical processing along withthe output of the last frequency comparator to logically identify whichof the four signals 1610 a-1610 d is faster.

FIG. 16 c shows a similar arrangement for the case of three inputsignals 1630 a-1630 c. In this fashion, one or more instances of thecircuits of FIGS. 16 a-16 c may be ganged in various combinations torealize a circuit implementation of the second cascading approach.

View of Square Wave Pair Dynamics as Symbolic Dynamics

Symbolic dynamics, also known as topological dynamics, is a rich area ofnonlinear science. Embodiments of the invention provide for symbolicdynamics [1-6] interpretations of the dynamics of a square wave pair.Such interpretations may be used as a design tool, as a means forcreating additional applications, and as a means for extendingfunctionality. In fact, the symbolic dynamics framework is an excellentsetting in which to define and explain many aspects of the invention.Only a very small amount of the symbolic dynamics formalism, andpractically none of the extensive theory, is necessary to obtain auseful engineering framework. This section presents the relevant,adapted, and interpreted material.

FIG. 17 shows one general setting in which to think of a symbolicdynamics system [6 (page 1)], although many others, often far moremathematically abstract, are commonly accepted [3,4,5]. In one version,a continuous-valued time, continuous-valued state dynamical system 1710has its state quantized, mapped to equivalence classes of states,projected onto a smaller collection of states, etc. 1720; all of theseresult in a small collection of discrete states (called symbols) whichare arranged in a (discrete-time) sequence 1730. The result is anoverall mapping of the continuous-valued time, continuous-valued statedynamical system 1710 to a discrete-valued time, discrete-valued state(a.k.a. symbols) dynamical system 1730. Equivalent discrete-valued time,discrete-valued state dynamical systems may model computers, machines,natural phenomena, etc., as well as equivalent abstract discrete-valuedtime, discrete-valued state dynamical systems which are purelymathematical. At a common level of abstraction, all of the equivalentforms may be thought of as identical. The study of the dynamics andother properties of such discrete-valued time, discrete-valued statedynamical systems is found in the subject of symbolic dynamics.

For the purposes of studying pairs of oscillators, some of the modelingmachinery of symbolic dynamics is helpful. FIGS. 18 a-18 c illustrate atorus model for the state-space of two continuous-valued oscillators.Here the periodic oscillations of one oscillator 1810 a-1810 c arerepresented as periodic motion in a vertical circle, while the periodicoscillations of second oscillator 1820 a-1820 c are represented asperiodic motion in a horizontal circle.

As shown in FIG. 18 a, the combinations of the circular patterns ingeneral sweep out the donut/bagel shape of torus 1830. If one associatesthe periodic timekeeping within each period of each oscillator with amoving point in the circular motion, and keeps track of this movingpoint for the combined pair of oscillators, the center of one circle iscentered at the location of the moving point of second circle 1825 inFIG. 18 a in a manner similar to adding the components of orthogonalvectors. This is illustrated in FIG. 18 b, and as the moving points ofeach oscillator actually move, a trajectory is traced out along thesurface of torus 1840 as illustrated in FIG. 18 c.

If one of the oscillators is faster than the other, it will wrap aroundthe surface of the torus faster in its rotation direction than will thatof the other oscillator. Note that all that is needed of thetorus/donut/bagel 1830 is its surface, and its interior is, for presentpurposes, hollow. In more formal terms this torus surface 1830represents a continuous-value state, continuous-valued time dualoscillator manifold, often used to describe or study uncoupled andcoupled linear and nonlinear differential equations and other systems.

In another view, shown in FIG. 19 a, a hollow tubular torus/donuttbagel1910 may be cut once in each of the two orthogonal directions andflattened out. Flattened torus surface 1902 comprises four edges 1905a-1905 d resulting from the two aforementioned cuts.

If edge A′ 1905 c is rejoined, or abstractly identified, with edge A1905 a and, similarly, edge B′ 1905 d is rejoined or abstractlyidentified with edge B 1905 b, torus 1910 can be readily reconstructed.

If both oscillators oscillated at precisely the same frequency andstarted in a lower corner of flattened torus 1925, the composite statetrajectory would move along diagonal line 1930 over time up to the faropposite corner. This traces out a connected winding path 1935 aroundthe torus 1940 comprising exactly one wrap in each direction, This isdepicted in FIG. 19 b. As time continues past the duration of oneperiod, the curve repeats, and the flattened torus corresponds tostarting over again at the original comber mentioned above.

If the two oscillators oscillate at different frequencies, morecomplicated windings occur. For example, if the oscillator correspondingto the horizontal direction oscillates at twice the frequency of theoscillator corresponding to the vertical direction, the torus will bewrapped with two turns in one direction and one turn in the otherdirection; the corresponding paths 1955 a, 1955 b on flattened torus1950 a, 1950 b appear as shown in FIG. 19 c. Similarly, if theoscillator corresponding to the horizontal direction oscillates at threetimes the frequency of the oscillator corresponding to the verticaldirection, the torus will be wrapped with two turns in one direction andone turn in the other direction; the corresponding paths 1970 a-1970 don the flattened torus 1960 a-1960 c appear as shown in FIG. 19 d. Thevarious copies 1950 a, 1950 b, and 1960 a-1960 c of the flattened torussurface depicted in FIGS. 19 c and 19 d may be lined up side-by-side,like flooring or wall tiles, and the trajectory can be connected to forman uninterrupted path. This “tiled” representation may be thought of asan unwrapped version of the wrapped trajectories on the surface of theunflattened torus.

The trajectories depicted in FIGS. 19 b-19 d show cases where the ratioof frequencies is exactly the ratio of two integers. This causes thetrajectory to always eventually meet back up where it started from andrepeat the pattern. The number of cycles involved for the repeatinvolves the least common multiple of the two frequencies. Moreformally, the trajectories depicted in the examples of FIGS. 19 b-19 dcorrespond to oscillator trajectories on manifolds for phase-locked,rational-valued frequency ratios of the two oscillators. In cases wherethe ratio of frequencies cannot be expressed as a ratio of integers(i.e., as a rational number), the trajectory path on the torus nevercrosses itself Additionally, FIGS. 19 c, 19 d illustrate how differentinteger frequency ratios between the two oscillators of FIGS. 18 a-18 cresult in differing wrapping characteristics and trajectory slopes.

This arrangement for abstracting the dynamic behavior of pairs of squarewaves can, in this way, be related to models of a dynamical system that,as a function of frequency ratio, produces either:

a regular (“self organizing”) pattern for so-called “commensurate”frequencies (i.e., the frequency ratio is a rational number), or

an irregular pattern (i.e., “quasi-periodic” or “chaotic” behavior) for“incommensurate” frequencies (i.e., the frequency ratio is not arational number).

These patterned behaviors of periodic binary waveforms for frequenciesvery close together, for frequencies related to these via aliasingphenomena, and/or waveforms comprising asymmetric pulses exhibit veryinteresting and complex attributes, yet are characterizable enough to beable to facilitate informative measurements with simple circuitry oralgorithms as seen thus far and in the material to follow.

FIG. 20 illustrates how the torus of FIGS. 18 a-18 c and FIGS. 19 a, 19b may be symmetrically quantized into regions associated with thesymbols employed by the invention, In particular, the symmetricquantization corresponding to the case where the input signals aresymmetric square waves is depicted. The square wave model is naturallyobtained by quantizing the continuous-value continuous-time oscillatortorus manifold 2010 into four regions 2020, 2030, 2040, and 2050corresponding to the symbols {S₀, S₁, S₂, S₃} as shown in FIG. 20.

Portions of trajectory paths on the torus surface (manifold) may then becharacterized according to which of the quantized sections they lie in,each corresponding to the symbols {S₀, S₁, S₂, S₃}. Referring to theleft side 2020, 2040 of torus 2010 of FIG. 20, oscillations occurringonly in horizontally-oriented loops (corresponding to, for example,oscillator A) alternate between symbols S₀ and S₂ while oscillationsoccurring only in horizontally-oriented loops on the right side 2030,2050 of torus 2010 alternate between symbols S₁ and S₃. Similarly,oscillations occurring in vertically oriented loops (corresponding to,for example, oscillator B) alternate between symbols S₀ and S₁ in thelower portion 2040, 2050 of torus 2010 and between symbols S₂ and S₃ inthe upper portion 2020, 2030 of torus 2010.

With both oscillators A and B active, the trajectories may then crossthrough all four sections 2020, 2030, 2040, 2050, generating anevent-driven symbol sequence having the symbols {S₀, S₁, S₂, S₃}. As atrajectory progresses from section to section, it may be thought of asgenerating an event-driven symbol sequence comprising one or more of thesymbols {S₀, S₁, S₂, S₃}) In this way, this embodiment of the inventioncan be viewed in the more formal symbolic/topological dynamics context.This model is referred to herein as the “quantized-symbol torus model.”

With this quantized-symbol torus model established, a correspondinginfinite periodic symbol tiling model is next developed.

In FIGS. 19 a, 19 b the surface of torus 1910, 1940 is represented as aflattened tile 1902, 1925 with edges 1905 a-1905 d, 1928 a-1928 d thatare identified together so that a trajectory crossing a location of oneedge reappears at the corresponding spot on the directly opposite edge.In FIGS. 19 c-19 d, a trajectory passing over the surface of the torusin differing ways was represented on a sequence of tiles 1950 a-1950 b,1960 a-1960 c. A general way to study complex wrappings of trajectorieson the surface of a torus is that of a “tiling,” wherein a sequence ofadjacent images of tiles, each corresponding to a full copy of thesurface of the torus, are arrayed edge-to-edge in a mosaic or grid.

In FIGS. 19 b-19 c, trajectories relating to oscillations with a fixedfrequency ratio appear as straight lines 1955 a, 1955 b, and 1970 a-1970d on each tile 1950 a, 1950 b, and 1960 a-1960 c. Edge A′ 1905 c isidentified with edge A l905 a of the same square and edge B′ 1905 d isidentified with edge B 1905 b of the same square 1902 to form torus1910. Alternatively, each edge A′ of a particular square may beidentified with an edge A of a different neighboring square to theright, and each edge B′ of a particular square may be identified with anedge B of a different neighboring square above. This forms a tiling(more precisely, an infinite periodic tiling). Neighboring tiles mayextend without bound in both vertical and horizontal directions, andthus trajectories relating to oscillations with a fixed frequency ratioappear as straight lines over the sequence of tiles.

Such an infinite periodic tiling may be further adapted to include thefour regions of the symmetrically quantized torus 2010 of FIG. 20. FIG.21 a illustrates a sequential tiling representation of the symmetricallyquantized torus 2010 of FIG. 20, wherein motion in the verticaldirection 2105 represents time and/or phase of one oscillator 2100 a,and motion in the horizontal direction 2110 represents time and/or phaseof the other oscillator 2100 b.

Each tile of the full torus surface is indicated with a thick lineboundary and is symmetrically subdivided into four separate areas. Thesesubdivided areas within each of the full torus-surface tiles will betermed “symbol tiles.” In the vertical direction, there are twoalternating types of columns, one that sequences between S₀ and S₂symbol tiles, and another that sequences between S₁ and S₃ symbol tiles.In the horizontal direction, there are two alternating types of rows,one that sequences between S₀ and S₁ symbol tiles and another thatsequences between S₂ and S₃ symbol tiles. This model will be termed the“infinite periodic quantized-symbol tiling model.” In this model, thearc-length of the trajectory corresponds linearly to measured time. Thusfor a trajectory with a slope of 2, oscillator A 2100 a changes symboltiles at twice that of oscillator B 2100 b.

With both oscillators A 2100 a and B 2100 b concurrently active, thetrajectories in this model may then cross through any of the symboltiles, and in this fashion may be seen as generating an event-drivensymbol sequence comprising the symbols {S₀, S₁, S₂, S₃} corresponding tothose generated as a trajectory wraps around the torus.

FIG. 21 b illustrates an exemplary trajectory on the sequential tilingrepresentation of FIG. 21 a. For visual simplification, the thickerboundaries have been equated to the other boundaries so that allboundaries are symbol tile boundaries. If the trajectory slope is notunity and the ratio of frequencies is not exactly that of two integers,symmetry events will occur in the sequence of crossings of symbol tiles,FIG. 21 b thus additionally depicts the identification of an exemplarysymmetry 2160 that is intrinsic to the exemplary trajectory.

As a side note, the boundaries of the symbol tiles are those of symbolboundaries; thus a symbol tiling represents an event-driven model. It isalso possible to create tilings corresponding to time-driven models, butsuch would amount to simply a histogram of the frequency ratio. Adiscrete-time time-driven model would also have to confront thecomplications of sampling at the “forbidden” state transitions where atrajectory simultaneously changes a symbol tile in both the vertical andhorizontal directions. In the physical world, the effects of temporalrace conditions, mechanical non-uniformity, and the like essentiallymale the “forbidden” state transitions a theoretical pathology.

Continuing the study made possible by the infinite periodicquantized-symbol tiling model, FIGS. 22 and 23 illustrate how varyingthe slope of a trajectory, the slope determined by the ratio of thefrequencies of the two oscillators, changes the class of incurredsymmetry events.

FIG. 22 shows a trajectory with slope greater than unity incurring a w₃₁symmetry 2210, as well as a trajectory with slope less than unityincurring a w₀₁ symmetry event 2220. Referring to the table of FIG. 3 b,the w₃₁ symmetry 2210 implies oscillator A 2205 a is faster than B 2205b, while the w₀₁ symmetry 2220 implies oscillator B 2205 b is fasterthan A 2205 a, in agreement with the situation depicted in FIG. 22.

In general, as the trajectory slope attains values above or below unity,differing classes of symmetry events occur. Each of the resulting twosymmetry event classes yields a complementary indication as to whichoscillator has the higher frequency. As to this, FIG. 23 illustratesexemplary portions of trajectories associated with each of the eightsymmetry events according to an embodiment.

Referring again to the table of FIG. 3 b, the four symmetry events w₀₂2315, w₁₃ 2325, w₂₀ 2310, W₃₁ 2320 incurred on trajectories with slopesgreater than unity are uniquely associated with oscillator A 2305 abeing faster than B 2305 b, while the four symmetry events w₀₁ 2345, w₁₀2340, w₂₃ 2335, w₃₂ 2330 incurred on trajectories with slopes less thanunity are uniquely associated with oscillator B 2305 b being faster thanA 2305 a.

FIG. 24 illustrates a larger scale view of a portion of a trajectory ofan exemplary pair of waveforms 2405 a, 2405 b whose relative ratio ofoscillating frequencies varies in some interval in time. Here, exemplarytrajectory 2410 varies through the infinite periodic tilingrepresentation as the ratio of the frequencies of the two oscillatorsystems 2405 a, 2405 b vary over time.

Several phenomena are depicted. Note the scale of the view isconsiderably larger, and each square represents an individual symboltile, here unmarked for the sale of clarity. First exemplary trajectory2410 experiences an epoch of time during which the two oscillators 2405a, 2405 b experience a fixed frequency ratio 2420, then an epoch with ameandering ratio 2430, and then an epoch of piecewise-constant periodicmodulation 2440. Other frequency-related trajectory phenomena,particularly those pertaining to frequency modulation and phasemodulation, and the like can be depicted, noting that frequencymodulation is time-differentiated phase modulation while phasemodulation is time-integrated frequency modulation.

Asymmetric Pulse Waves and Their Symbolic Dynamics Phenomena

Many common oscillators constructed from feedback loops introducedaround logic gates (for example, CMOS inverter gates) produce slightlyasymmetric waveforms which differ in duty-cycle from that of completesymmetry (50%) by values such as 3% or more. In practice, all squarewave oscillators will produce binary-valued waveforms that are at leastslightly asymmetric.

Additionally, when the frequency of a square wave oscillator ismodulated, asymmetries in pulse width are introduced as the waveformperiod expands or contracts. If the frequency is modulated rapidly withrespect to the oscillator frequency or with large modulation index,these asymmetries can be significant. If the frequency is modulatedrelatively slowly with respect to the oscillator frequency or with alarge modulation index, the asymmetries are insignificant.

When an asymmetric input binary-valued pulse waveform is sufficientlyclose in frequency to that of another waveform to which it is compared,a number of additional phenomena can occur:

the narrower portion of one asymmetric square wave experiences anenveloping event with respect to either:

-   -   symmetric portions of a fully symmetric ideal compared square        wave, or    -   the wider portion of the other compared asymmetric square wave;

the wider portion of one asymmetric square wave experiences anenveloping event with respect to either:

-   -   symmetric portions of a fully symmetric ideal compared square        wave, or    -   the narrower portion of the other compared asymmetric square        wave.        These additional phenomena can confuse the methods described        thus far.

The degree to which this situation can even occur is bounded by howclose the waveforms are in frequency and how asymmetric each of thewaveforms is. In cases where the frequencies of two compared waveformsare close enough to create these phenomena, a natural solution for manyapplications is to employ toggle flip-flops to homogenize the symmetryas described earlier in conjunction with FIG. 5 c. However, varioustypes of asymmetric conditions and the phenomena they induce can bereadily detected, and in many cases adverse effects may be readilycorrected. This section considers extensions that handle asymmetricbinary-valued pulse waveforms.

To begin, FIGS. 25 a-25 d illustrate exemplary phenomena pertaining tothe symbolic dynamics of asymmetric binary-valued pulse waveforms. Suchinput waveform asymmetry, when applied to implementations of theinvention designed specifically for symmetric input waveforms, cancreate problematic alternating indications that each of the oscillatorsis faster than the other when the two oscillator frequencies aresufficiently close together.

Referring to these figures, four cases (polarity combinations) ofasymmetric pulse waves of almost the same frequency and almost the same(asymmetric) duty cycle 2510 a-2510 d are shown. As may be seen in thesefigures, asymmetric pulse waveforms have unique signature events aswell, expressible in terms of “signature” sequences of symmetry events:if the enveloping event resulting from the asymmetry involves briefflashes of the S₀ symbol 2515 a-2515 c, there will be a sequence of w₀₁2525 and w₀₂ 2520 symmetry events separated by one non-symmetry 2517,2519.

If the enveloping event resulting from the asymmetry involves briefflashes of the S₁ symbol 2525 a-2525 c, there will be a sequence of w₁₀2535 and W₁₃ 2530 symmetry events separated by one non-symmetry 2540,2545.

If the enveloping event resulting from the asymmetry involves briefflashes of the S₂ symbol 2550 a-2550 c, there will be a sequence of w₂₀2560 and W₂₃ 2570 symmetry events separated by one non-symmetry 2565,2567.

If the enveloping event resulting from the asymmetry involves briefflashes of the S₃ symbol 2575 a-2572 c, there will be a sequence of w₃₁2580 and w₃₂ 2593 symmetry events separated by one non-symmetry 2590,2596.

Note the pairs of symmetry events in each list item are complementary inthat the first symmetry event implies one oscillator would be faster ifits pulse waveform were symmetric, while the other symmetry eventimplies the opposite. Without special considerations, then, thesephenomena create problematic alternating indications that each of theoscillators is faster when the frequencies are close enough that slightasymmetries in the input waveforms are within the measurement capturerange of the aforementioned implementations.

Several additional interesting algebraic relationships are also inherentin the above list of observations. Additionally, if the frequencies ofthe two square waves are very close, the sequence will repeat in analternating pattern for two or more times, the number of timesincreasing monotonically as the frequencies become even closer to beingidentical. Further, each of the conditions in the above listindividually identifies a particular type of relative asymmetry inherentin the pair of waveforms. These facts, and similar ones resulting fromother cases (more divergent asymmetry, one waveform essentiallysymmetric while the other is asymmetric, etc.) may be used to createadditional circuitry or algorithms to isolate and indicate these events,unique to the asymmetries involved. Additionally, once tile asymmetriesare detected, the phenomena that would otherwise produce a con-fusedoutcome for the circuitry and algorithms can be used to producedefinitive frequency comparison outcomes, and even provide additionalinformation characterizing categorical details of the asymmetries.

An example is provided in FIG. 26, which illustrates an exemplaryadaptation made to the symbol-based embodiment of FIGS. 8 a-c to handleasymmetric pulse waveforms. Following from the list provided above, afirst pair of RS latches 2620 a, 2620 b are configured to set when asequence of w₀₁ 2610 a and w₀₂ 2610 b symmetry events, respectively, areobserved and reset when another symmetry event is observed. Notesymmetry events w₀₁ 2610 a and w₀₂ 2610 b are complementary in that w₀₁2610 a implies oscillator B would be faster if its pulse waveform weresymmetric while w₀₂ 2610 b implies oscillator A would be faster if itspulse waveform were symmetric; thus these events would not occur in thissequence if the two input pulse waveforms were symmetric and theirfrequencies were in a fixed ratio.

By taking a logical AND of these two captured events, the firstasymmetry signature in the list above would be subsequently indicated.Using NAND gate 2630 a to realize this AND operation facilitates use ofan additional NAND gate 2635 a to provide (via DeMorgan's law) an ORoperation as well as drive an LED indication 2640 a of the S₀ asymmetryevent.

Similarly, three additional pairs of RS latches 2620 c-2620 d, 2620 e,2620 f, 2620 g, and 2620 h are responsive to the remaining t-ree pairsof signature symmetry event sequences 2610 c, 2610 d, 2610 e, 2610 f,2610 g, and 2610 h, each similarly directed to a corresponding NAND gate2635 b-2635 d and LED 2640 b-2640 d in the same fashion. All four of theS₀, S₁, S₂, S₃ asymmetry events are thus separately captured, separatelyprovided with LED indication, and directed to an OR operation toglobally report the general existence of an asymmetry event.

The general background as to the behavior of pairs of asymmetric pulsewaveforms can be further rendered in terms of modified versions of thesymbolic dynamics models introduced for the symmetric case. As to this,FIG. 27 illustrates a variation of the symmetrically quantized torus2010 of FIG. 20 adapted for use with asymmetric pulse waveforms. Here,region boundaries 2710 a, 2710 b determining the quantization thresholdson the continuous-value continuous-time oscillator torus surface may beadjusted (indicated by arrows 2720 a, 2720 b) to obtain different dutycycles. Clearly this affects the resulting event-driven symbol sequence.

Further detail can be seen by adapting the infinite periodicquantized-symbol tiling model to asymmetric pulse waveforms. As to this,FIG. 28 a illustrates an asymmetric sequential tiling representation ofthe asymmetrically quantized torus 2705 of FIG. 27, akin to thesymmetrically-quantized torus of FIG. 21 a. As with FIG. 21 a, eachlarger tile corresponding to the full torus surface is indicated with athick line boundary, and each larger tile of the full torus surface issymmetrically subdivided into four separate areas. These subdividedareas within each of the full torus-surface tiles will be termed “symboltiles.”

In the vertical direction 2820 a, there are two alternating types ofcolumns, one that sequences between S₀ and S₂ symbol tiles and anotherthat sequences between S₁ and S₃ symbol tiles. In the horizontaldirection 2820 b, there are two alternating types of rows, one thatsequences between S₀ and S₁ symbol tiles and another that sequencesbetween S₂ and S₃ symbol tiles.

FIG. 28 b illustrates three exemplary trajectories associated with fixedfrequency ratios of unity 2850 b, less than unity 2850 c, and greaterthan unity 2850 a in their traversal over the asymmetric sequentialtiling. Using the notions of FIGS. 28 a, 28 b, FIG. 28 c illustratesexemplary portions of trajectories associated with each of the eightsymmetry events of the invention. Note in particular that the twoparallel solid-marked spans of symmetry events w₂₃ 2870 and w₀₁ 2875 areof equal length, while at a slightly different frequency ratio the twodashed-marked spans of symmetry events w₂₃ 2880 and w₀₁ 2885 are ofunequal length, resultant from the waveform asymmetry.

Attention now is directed to the relation between closeness in frequencyand degrees of asymmetry that give rise to asymmetric events and theduration of constituent alternating patterns of complementary eventsymbols.

To begin, FIG. 29 a illustrates timing notations that may be applied toan exemplary pair of asymmetric input waveforms that share a nearlyidentical asymmetric duty cycle. The top waveform comprises anasymmetric pulse pattern with a given duty cycle (other than 50%) whichrepeats periodically at a given rate to form pulse-wave A 2910 a. Thelower waveform is a time-stretched version of the original asymmetricpulse pattern; this time-stretched version repeats periodically to forma slower pulse-wave B 2910 b of lower frequency of the same duty-cycle.

In this example the period of the time-stretched version is depicted as7/5 longer than the period of the original asymmetric pulse pattern.Herein, the ratio of frequency of pulse wave A 2910 a to frequency ofpulse wave B 2910 b is 7/5, and their co-aligned phase-locked patternrepeats every 7 cycles of pulse wave 2910 a and every 5 cycles of pulsewave B 2910 b. Note that for the predominant cases where one or both ofthe frequencies either drift from the illustrative phase-lock conditionof FIG. 29 a or are non-commensurable, the relative positions of thewaveforms will shift variably over time.

For subsequent discussion and additional calculation, the followingquantities are in general defined:

the period of waveform A 2910 a is T_(A) 2920 a, measured in units oftime;

the period of waveform B 2910 b is T_(B) 2920 b, measured in units oftime;

the duty-duration of waveform A 2910 a is δ_(A) 2930 a, measured inunits of time;

the duty-duration of waveform B 2910 b is δ_(B) 2930 b, measured inunits of time;

the duty-cycle of waveform A 2910 a is α_(A) 2940 a, measured asdimensionless fraction;

the duty-cycle of waveform B 2910 b is α_(B) 2940 b, measured asdimensionless fraction;

the duration over which waveform A 2910 a has logical value 0 isT_((a=0)), measured in units of time;

the duration over which waveform A 2910 a has logical value 1 isT_((a=1)), measured in units of time;

the duration over which waveform B 2910 b has logical value 0 isT_((b=0)), measured in units of time;

the duration over which waveform B 2910 b has logical value 1 isT_((b=1)), measured in units of time;

the period ratio of waveform B 2910 b to waveform A 2910 a isβ=T_(B)/T_(A).

Note that δ_(A) 2930 a and T_((a=1)) are equivalent, as are δ_(B) 2930 band T_((b=1)). Additionally, for the example above, β= 7/5 andα_(A)=δ_(B). Also since period is the reciprocal of frequency, theperiod ratio of waveform B to waveform A is identical to the frequencyratio as defined throughout this specification.

In a situation in contrast to that of FIG. 29 a, FIG. 29 b illustrates aportion of a case where the frequency ratio is only slightly larger than7/5. This frequency ratio guarantees the eventuality of a situationdepicted in FIG. 29 b wherein the evolving occultations of theasymmetric aspects of two exemplary asymmetric input waveforms 2950 a,2950 b with frequencies sufficiently close together result inalternating indications 2960 a-2960 e and 2970 a-2970 e that each of theoscillators 2950 a, 2950 b in turn is faster than the other.

In the slightly larger view, pairs of asymmetric waveforms atsufficiently close frequencies experience alternating intervals ofgeneral typical behavior and intervals of asymmetry events comprisingalternating indications of symmetry events (the situation depicted inFIG. 29 b being an example of the latter) as non-phase-locked waveformsslip by one another. This alternation can be viewed as a type ofmacro-cycle (although if the frequencies are non-commensurable, thismacroscopic behavior will not be periodic but rather slowly evolving).

In general, the macroscopic behavior will be similar to that depicted inFIGS. 30 a-30 d. In particular, FIGS. 30 a-30 d depict four macroscopicbehavioral signatures, respectively, corresponding to the asymmetryevents depicted in FIGS. 25 a-25 d. These four signatures aredistinguished by the presence of specific pairs of complementarysymmetry events. Detection of these signatures, further, specificallyindicates relative relationships that must exist among T_((a=0)),T_((a=1)), T_((b=0)), and T_((b=1)). Note that all of this is simply aformalization of the geometric pulse width relationships depicted inFIGS. 25 a-25 d.

In a more aggregate and general behavioral view of the macro-cyclesituations depicted in FIGS. 30 a-30 d, FIGS. 31 a-31 c depict theevolution of macro-cycle behavior that occurs for waveforms comprising(even minor) asymmetry as the frequency ratio is increased from a valuesufficiently lower than unity, through ratios sufficiently close tounity, and then in ratios sufficiently greater than unity.

FIG. 31 a illustrates macro-cycle behavior for the case where thefrequency ratio is a value sufficiently lower than unity. FIG. 31 billustrates the case where the frequency ratio is close enough to unitythat the waveform asymmetry (however small it may be) becomestheoretically detectable. However, in any physical implementation, theminimum response times of the logic circuitry, mechanical apparatus,chemical process, sampling rate for data provided to algorithmicimplementations, etc. will determine a minimum threshold for which thissituation can be detected. As the frequency ratio continues to increaseto ratios sufficiently greater than unity, the behavior will become thatdepicted in FIG. 31 c.

The process giving rise to the phenomena of FIG. 3 1 b may be understoodin terms of the example of FIGS. 32 a-32 c. For two waveforms whosefrequencies are sufficiently close together, one of the waveforms may beviewed as progressing past the viewpoint of the other waveform. The rateof this progression is proportional to the difference in frequencies(and as such is similar to a “beat frequency,” well-laIown in thepractices of vibrational mechanics, acoustics, and communications). Asone waveform progresses past the other, the waveform asymmetries undergovarious enveloping arrangements (or occultations) of one another.

In particular, FIGS. 32 a-32 c illustrate the evolution of symbolsequences and symmetry events before, during, and after occultations ofasymmetric aspects of two exemplary asymmetric input waveforms withfrequencies sufficiently close together. Before occultation, there areno symmetry events, as illustrated in the specific illustrative exampleof FIG. 32 a. During occultation, the asymmetric aspects give rise tovarious asymmetry events, in particular the sequence of complementarysymmetry events 3240, 3245, 3250, 3255, as illustrated in FIG. 32 b,depicting the slightly shifted epoch in specific illustrative example ofFIG. 32 a. After occultation, the asymmetric aspects no longer give riseto asymmetry events and the situation returns to one not unlike thesituation of FIG. 32 a. This is illustrated in FIG. 32 c, depicting ayet further shifted epoch in the specific illustrative example of FIG.32 a.

Formulas may be computed, in terms of the quantities defined earlier,for the upper and lower thresholds at which, for a given frequency ratioand given duty cycles, the behavior of FIGS. 31 b will begin to appear.These formulas can be extended to further characterize thresholds formore detailed behavior such as that depicted in FIGS. 30 a-30 d, thetemporal duration of macro-cycles, the number of alternations ofcomplementary symmetry events in a macro-cycle and the frequency ofthese alterations, and the like. Threshold formulas may be Furtheradjusted to account for minimum response times, sampling rates, andother factors to determine a minimum threshold for which asymmetrysituations can be detected. Such formulas can be used to characterizethe limitations of a particular embodiment. Additionally, such formulasmay be used to design or create additional feattres, capabilities, andextensions.

For example, the use of resettable counters in conjunction withdetection of specific alternating complementary symmetry events may beused to further characterize waveform asymmetries and specific aspectsof asymmetry events that may be useful in specific applications.

The various methods and processes described herein may be implemented ina computer-readable medium using, for example, computer software,hardware, or some combination thereof. For a hardware implementation,the embodiments described herein may be performed by the above-notedcircuitry and/or in conjunction with a processor. As such, hardwareembodiments may be implemented within, for example, one or moreapplication specific integrated circuits (ASICs), digital signalprocessors (DSPs), digital signal processing devices (DSPDs),programmable logic devices (PLDs), field programmable gate arrays(FPGAs), processors, controllers, micro-controllers, microprocessors,other electronic units designed to perform the functions describedherein, or a selective combination thereof.

For a software implementation, the embodiments described herein may beimplemented with separate software modules, such as procedures,functions, and the like, each of which perform one or more of thefunctions and operations described herein. The software codes can beimplemented with a software application written in any suitableprogramming language and may be stored in a memory unit, and executed bya processor. The memory unit may be implemented using any type (orcombination) of suitable volatile and non-volatile memory or storagedevices including random access memory (RAM), static random accessmemory (SRAM), electrically erasable programmable read-only memory(EEPROM), erasable programmable read-only memory (EPROM), programmableread-only memory (PROM), read-only memory (ROM), magnetic memory, flashmemory, magnetic or optical disk, or other similar or effective memoryor data storage device.

Other Applications

It was noted earlier that realizations of the invention can also be usedto determine which of two buttons or switches was last cycled on andoff. Additional logic circuitry can be added to create a “radio-button”chip that can be ganged to arbitrary numbers of buttons. Further, theinvention may be adapted to a wide range of phase-detectionapplications.

The invention is applicable to a wide range of possible applicationareas. These include both system and method realizations, as well asrealizations of analytical methods, Some examples of applicableapplication areas include:

synchronous motor control and operation;

cross-traffic or resource-constrained timing in transportation systems;

resource allocation in manufacturing or project scheduling theory;

scheduling of real-time tasks in real-time and near-real-time operatingsystems;

astronomy calculations, perhaps applied in new analyses of ancientarchaeoastronomy sites;

tool for study of oscillator-coupling phenomena in chaotic andself-organizing systems;

geometric lattice design;

quantum effects in nanotechnologies;

long-duration timing systems as may be used in radioactive waste storageor long-distance space travel;

study of molecular vibration;

study of energy-transfer among inharmonic periodic modes of oscillation;

analysis of biological and ecological systems.

Those skilled in the art will realize the wide range of alternativesapplicable to embodiments of the invention; these are recognized due tothe general character of the invention and are thus provided as part ofby the invention.

While the invention has been described in detail with reference todisclosed embodiments, various modifications within the scope of theinvention will be apparent to those of ordinary skill in thistechnological field. It is to be appreciated that features describedwith respect to one embodiment typically may be applied to otherembodiments. Therefore, the invention properly is to be construed onlywith reference to the claims.

REFERENCES

[1] Bai-Lin, Hao, Elementary Symbolic Dynamics and Chaos in DissipativeSystems, World Scientific, Singapore 1989, ISBN 9971-50-682-3.

[2] Hale, Jack K.; Kocak, Huseyin, Dynamics and Bifircations,Springer-Verlag, New York, 1991, ISBN 0-387-97141-6.

[3] Kitchens, Bruce P., Symbolic Dynamics—One-sided, Two-sided andCountable State Markov Shifts, Springer-Verlag, Berlin 1998, ISBN3-540-62738-3.

[4] Nerurkar, M. G.; Doldcen, D. P.; Ellis, D. B. (eds.), TopologicalDynamics and Applications, Contemporary Mathematics, AmericanMathematical Society, Providence 1998, ISBN 0-8218-0608-4.

[5] Walters, Peter (ed.), Symbolic Dynamics and its Applications,Contemporary Mathematics, American Mathematical Society, Rhode Island1992, ISBN 0-8218-5146-2.

[6] Williams, Susan G. (ed.), Symbolic Dynamics and its Applications,Proceedings of Symposia in Applied Mathematics, American MathematicalSociety, Providence 2004, ISBN 0-8218-3157-7.

1. A method for comparing frequency, the method comprising: detectingtransitions of a first input waveform; detecting transitions of a secondinput waveform; wherein if a plurality of transitions of one waveformoccur between two consecutive transitions of the other waveform, saidwaveform possessing said plurality of transitions is identified as thefaster of the first input waveform and the second input waveform.